$5,500 for business class transpacific flight during spring break seems really reasonable. I don't know what you are expecting? LAX-TPE and SFO-TPE have high load factor and lots of business travelers paying premium fare. Have you try premium economy? I think that is more in your price range.

A couple of things you can try:
1. Fly out on Tuesday or Wednesday instead of the start or end of the week. Fares tends to be cheaper mid week.
2. Look for back track fares... try BR LAX-IAH-TPE, LAX-ORD-TPE or LAX-SEA-TPE. You may have to book this from a partner airline.
3. Look for hidden city fares... try LAX-TPE-HKG or LAX-TPE-NRT on CI or BR and extend the layover at TPE by a week.
4. Buy full fare economy or premium economy and use your miles for upgrade.
5. Look for 2 seperate one-way tickets.
6. Look for open jaw fares... try LAX-TPE // NRT-LAX or LAX-HKG // TPE-LAX and just buy an one-way discount carrier ticket for the missing link.
7. Have you try looking for fares to TSA instead of TPE? This is a long shot but maybe there is some wacky LAX-HND-TSA specials that happens to be open on the dates your parents want to travel. You can combine this one with the #5 and #6 above. You may be surprised.
8. Use Amex Platinum travel website, which has discount business class fare not available from the airlines. You have to get a Amex Platinum card of course but it may be worth the first year fee.
9. Just pay $5,500. It's not that bad.