If only you were going to Singapore, then it wouldn't be a bad idea to wait for your parents at the airport since there is lots to do at Changi Airport...such as free cinema, swimming pool, many areas to relax, all airside...but alas, Suvarnabhumi (BKK) has no such facilities for arriving pax. In my view, Thailand really blew it (Given that Suvarnabhumi is brand new) to build a new facility that cannot compete with Changi in Singapore for amenities. If transiting in Southeast Asia, Changi cannot be beat. Given that Both BKK and SIN are travel hubs, I think many travelers route their itineraries through SIN due to the superior facilities at Changi airport.

If I were you I would simply take a taxi to your hotel and have your parents do the same. Its only about a 30 minute taxi ride to central Bangkok (if you use the toll expressways, you pay the toll). Just go to the public taxi stands on level 1. If you use a public taxi from the aiport, you will pay the meter rate, plus 50 baht, plus whatever expressway tolls are paid. You will likely pay between 300 and 400 baht in total. I do not recommend using the private taxi touts who swarm you as soon as you emerge from customs, Ignore them and go down to level 1 where the public taxi stand is.

Also, if you don't have Thai currency upon arrival, you can exchange cash at many points in the airport. I usually use the ATM which is in the customs hall, just after passing immigration.