OP - A suggestion that you take a step back. Before discarding the train option, discuss this with a TA who specializes in the region and perhaps travel for elderly individuals. The specific stations and trains will matter, but you will not exactly be travelling on the NY Subway and, depending on the extent of your parent's needs, they may well be quite comfortable with the acommodations. Perhaps moreso than you can provide in a vehicle with which you are unfamiliar on roads with which you are unfamiliar and with rules with which you are unfamiliar.

This is the kind of thing which really is detail oriented and, if you are not intimately familiar with the details of each station, the services available, and exactly what their needs are, it is well worth paying a fee to learn from someone who provides that sort of advice every day. Especially if you are averse to driving in the first place, have logistical challenges and might be able to avoid all of it.