I run a hotel in the Colonial Zone, so I have a vested interest. But I honestly think there is really only one option, and that is to book a hotel in the Colonial Zone, and get the hotel to arrange an airport pick-up. The airport has had a lot of problems during the night, and Santo Domingo generally is having a crime wave. The next day, the only feasible bus to put your parents on to Las Terrenas is the 1.40pm departure, it leaves from near to Duarte and they would have to go there to pick up the bus to make sure they get seats together. All the other buses are the 25/30 seaters, and they cram people in. The 1.40pm departure has air conditioning and is a proper sized bus. So, they will get a safe first night, be able to have breakfast leisurely in the morning then take off at about 12 midday after check-out and be at the bus station to pick up the bus.
Colonial Zone hotels are almost certainly the best value on the island. Mine is 4 star and we are currently offering a rate for 2 including breakfast, Wifi, air conditioning, use of a gym, a pool, jacuzzi of just US$ 69 plus tax. I'm happy to give you more information if you PM me.
I would seriously advise against trying to get your parents from Santo Domingo to Las Terrenas during the night, even in a private taxi. The road is quite dangerous at night, and there are a lot of cattle straying onto it, as well as treacherous bends.
We all like to save money, but honestly, please take my advice and play this one safe. If anything happened to your family it would ruin their holiday and you would never forgive yourself. Remember, when they arrive, they will be at their most vulnerable as they will have their passports, their money and all their valuables with them.