mariposita is correct. While Hostal Adriano
is a very very nice place, well appointed, refrigerator in every colorful room, those 5 flights of stairs can be challenging if your parents aren't in good shape - and it doesn't look like the elevator will be finished by then either. Otherwise, Hostal Adriano is very very well located in the center of it all. Plus, they have free internet access if your parents will be taking their own laptop. Hostal Martín
is a good choice too, with simpler rooms, elevator in the building, very clean, streetside rooms have balconies, and the building is a one-time palace (or at least a house of royals). I spent a night in the "sister-hostel" of Hostal Cervelo
, located on the next floor and owned by the same family, and found it to be quite nice.
But your question is a good one. I guess it depends on the expectations of your parents, their age, their health, etcetera. I think many older people tend to have higher standards and/or are accustomed to USA Hotels which are generally rather nice, spacious, well decorated, etcetera. Hostels (called "hostales" in Spain - and shouldn't be confused with YOUTH hostels which are called "albergues juveniles") in Spain TEND to have small rooms with a tiny bathrooms, thinner walls, don't tend to have bellhops or anyone to carry baggage up stairs (but they do it for you at Hostal Adriano, I know), there's not always someone at the front desk, sometimes entryways can be dark(er), and hostels are almost always located UP to the next levels of buildings.
You'll probably pay about half of what you'd pay for a hotel though. Older people tend to require a lot more to make them comfortable and keep them happy - no matter the price. Some people stay in hostels and expect to be treated like kings and queens and this just isn't going to happen with hardly any hostels - although some do come close.
Best of luck! Saludos, MadridMan