The Alhambra complex in Granada covers a lot of ground and is on the side of a hill. The mobility-challenged might want to limit their visit to the Nasrid Palaces, the most spectacular part of the complex, and perhaps the nearby Charles V Palace (museum). There is bus service up to the edge of the complex. I don't know whether a taxi would get you closer. Attempting to see the whole thing would take more than half a day.
Parc Guell in Barcelona requires some walking, and the entrance is quite some distance from the nearest subway stop. I don't know whether you can get closer by bus; a taxi might be smarter. I believe taxis in Barcelona are reasonably priced.
What time of year do you plan to travel? The heat in much of Spain--but especially in Andalusia--is often really brutal in the summer. That can be an issue for anyone.
The only place a rental car would help you would be the white villages in Andalusia. As already mentioned, they may be hilly enough to be a problem for your group. If they are not--or if you're willing to spend some time in a car in order to get the picturesque views from a distance, you might consider renting a car just for the drive between Seville and Granada. It's really not very practical to see a bunch of the villages by public transportation unless you have a lot of time and are willing to move from small city to small city, seeing one village from each base. [I see the last comment was inaccurate in its reference to "small city"; I meant to say that you can spend at least one night (might need two) in Ronda and take a day trip to Grazalema from there, and you can spend a night in Jerez for a day trip to Arcos de la Frontera. Arcos is doable from Seville but that requires a lot more time on buses than it does from Jerez. I'm not sure where you'd need to be based to accomplish a day trip to Zahara de la Sierra.]
