(bolding mine)

Not sure about the award availability in May.

Just a point of reference so you may have a reasonable expectation:

An award tix of 60K miles for SFO-TPE in Y was quite standard and had been for years. I am talking about pre-2010 using AA, UA, NWA miles.

A C award normally cost 90K to 120K miles. Maybe someone in the know can chime in but personally I've never seen any C award with 60K miles for many many years. I recall a (big) limited time promo from UA back in 2004 or something that required only 50% of miles when flying to Asia (Japan, Taiwan..etc).

As for your options, currently without any transfer, you don't have enough UA miles for a tix in Y. One-way, yes; but not RT. The UA route with Japan connection is also quite normal. IIRC, there's no direct flight between SFO and TPE no more since SARS scares. If I am not mistaken, it'll cost more if using AA miles flying to TPE.

Also keep in mind May is just at the corner and the award availability only decreases as time goes by, even for Y awards. I think the high season starts late May and for the rest of summer. Meaning award tickets are harder to come by without advanced planning. If I were you and would like to make it to Dad's B-Day, I would grab whatever is available right now.

Good luck!