There are two types of benefit for which you may be eligible when a family member dies while you're working in Taiwan.
1) Bereavement leave (喪假) aka funeral leave
For most working people, this is covered by Article 43 of the Labor Standards Act (勞動基準法), which refers to Article 3 of the Regulations of Leave-Taking of Workers (勞工請假規則). The number of days you're allowed to take off depends on the type of relationship (not strictly the degree of kinship) and ranges from 8 days for the death of a parent to 0 for a cousin, a great-great-grandparent, or any other relative considered too distant. The text of Article 3 covers some questions (e.g. step-parents and foster parents are included), but other situations depend on interpretative letters from the Ministry of Labor or its predecessors such as the CLA (e.g. a parent of your ex-spouse doesn't count, but the parent of your deceased spouse does) or may still be unanswered.
This is paid leave. If you have a fixed monthly salary, it should be easy to calculate because you just take the relevant number of days off and receive your full salary as usual. Otherwise, you probably need to use the formula in 6.3.4 of the 僱用部分時間工作勞工應行注意事項 (only on the MOL's website, and no official English version available). Presumably the formula should be revised to reflect the 40 hour work week now being standard, but for now this is what it says:
(number of hours you're scheduled to work in a 2 week period divided by 2) divided by 42 multiplied by 8 multiplied by number of days you're entitled to take off
As with holiday pay, there's a different formula for employees paid by "piecework", based on the "average wage" formula in the Labor Standards Act. Hourly pay is not considered piecework. Your payment for the leave period counts as part of your "wage" or salary and is not legally a "bonus". It is taxable.
You do not need to attend the funeral (this has been the subject of an official interpretation). There is no firm deadline by which you need to apply for the leave, but the custom is within 100 days of the death. The leave period can be divided into two parts if "required by custom". Your employer can ask for "relevant documentary evidence", the scope of which is not clearly defined.
If your employer fails to include bereavement leave in your contract or work rules (or fails to provide you with a copy of the work rules) and does not inform you that you have this right, you have a reasonable claim for compensation even after the 100 days (but still limited to the number of years prescribed for such claims in the Civil Code). An employer that violates Article 43 of the LSA is subject to a minimum fine of $20,000 via Article 79, though it takes some time for this to be enforced if the company puts up a fight.
If you are among the 6% or so of the workforce not covered by the Labor Standards Act, your entitlement may be different (or non-existent). "Part-time" buxiban teachers are definitely covered.
2) "Death benefit" (喪葬津貼) aka funeral benefit
This is a separate issue and has no direct connection to the Labor Standards Act; this one depends on the Labor Insurance Act (勞工保險條例) instead. Most employees should have labor insurance (勞工保險 or 勞保 "laobao"). This is different from employment insurance (就業保險), which doesn't apply to you if you're a foreigner without a local spouse.
The "death benefit" in the Labor Insurance Act (勞工保險條例) is not limited to your own death. Article 62 covers family members of the "insured person" (that's you), but the range of family members covered is narrower than that of bereavement leave. Only your parent, your spouse, or your child counts. The interpretations regarding step-parents etc. may also be different. If the child dies in a miscarriage, this benefit is not applicable (it's separate from maternity leave and sick leave). This benefit does not give you time off, but it does give you 1.5, 2.5, or 3 times your monthly salary, depending on who died and the age of the child if applicable.
To make things extra confusing, the method for calculating your salary in this case is different from the "average wage" formula in the LSA. Basically,
LSA average wage formula: all "wages" applicable for work performed in the previous 6 months (ending the day before the event occurs) divided by 6
LIA "average insured salary" formula: all "wages" applicable for work performed in the previous 6 months (including the day on which the event occurs) divided by 6, but subject to salary grades including minimum and maximum grades (currently $20,008* and $43,900 respectively, subject to revision each year)
*lower for part-time workers and some other cases
As stated in 8.3 of the 僱用部分時間工作勞工應行注意事項, if you had more than one job in the relevant 6 month period for which you were or by law should have been insured, the total combined salary is applicable, still subject to the salary grades. (There are some ambiguities about exactly how to calculate it, which I've asked the Bureau of Labor Insurance to clarify, so far without much success.)
If your employer under-reports your salary (both you and your employer have to pay premiums every month, so the employer has an incentive to under-report) or reports the actual amount paid but should have paid you more, the amount that you should have been paid and that should have been reported is what counts in the end.
If you're insured at the time of the death, you claim this benefit from the government, not your employer. If you're not insured but should have been, you need to sue the company (unless it can be persuaded to pay you), on the basis of Article 184 of the Civil Code. Some people have optional labor insurance, but for most people it's mandatory, and "part-time" buxiban teachers are definitely included.
An employer that fails to register you for insurance or over- or under-reports your salary is subject to a penalty of 4 times the missing or excess premiums. An employee who fails to "participate" in labor insurance can also be fined, but the maximum fine is $500.
The standard of proof for this benefit is defined in Articles 54 and 82 of the Enforcement Rules of the Labor Insurance Act (勞工保險條例施行細則). If the death occurs in a foreign country, the death certificate needs to be authenticated by the ROC embassy or quasi-embassy (TECO, TECRO, etc.). As the various types of "death certificate" issued by governments, hospitals, and funeral parlors in different jurisdictions don't necessarily conform to Taiwan's standards, the exact type of document needed should be confirmed with the Bureau and the TECO etc. A Chinese translation is required, which also needs to be authenticated. There seems to be a discrepancy between the Chinese and English versions of the paragraph that mentions this (ironically enough), regarding whether the translations need to be produced by a TECO etc. or just authenticated by one.
Edit: The "household registration form" listed in Article 82 (to prove the family relationship) can pose a problem when a parent or child dies in a country that doesn't have an equivalent to the hukou system. A marriage certificate can prove a spousal relationship, and a birth certificate listing the child's parents can prove a parent-child relationship. Otherwise, ask the Bureau (or the relevant court if the Bureau isn't involved) exactly what you need. As for translations, I've been told by a court here that although the original documents need to be authenticated by the TECO/embassy/etc. in the country that issued them, the translations can be produced by anyone and then authenticated in Taiwan by showing them to a court's notary department, along with the originals.
The government's deadline for claiming the labor insurance death benefit is 5 years.
The Bureau's website: http://www.bli.gov.tw
MOJ's partly bilingual law database: http://law.moj.gov.tw
MOL's Chinese law and interpretation database: http://law.mol.gov.tw
MOL's English law database: http://laws.mol.gov.tw/Eng/Default.asp (because the English link in the Chinese version only says "英文" not "English" )
Ps. Before the chorus starts, let me say I do agree that asking your employer to follow the law can result in unfavorable actions towards you, as discussed in many, many other threads here, so if you find yourself in this situation, please take note of the deadlines and consider a timetable for taking whatever action you deem appropriate. If your boss breaks the law in a way that gives you a reasonable fear of your rights and interests as an employee being violated, the deadline to quit (and receive severance pay) is 30 days from your discovery of the violation. If you don't get paid correctly, this deadline does not apply (Labor Standards Act, Article 14).
Pss. All labor insurance benefits are tax-free (Income Tax Act Art. 4 Par. 1 Subpar. 7). If you ask the Bureau of Labor Insurance to investigate suspected over- or under-reporting of income, misappropriation of premiums, and/or failure to register for insurance, you can do so confidentially, but (as with other labor inspections) the investigation will be more thorough if you allow yourself to be identified to the company.