Just looking over that Ron In Rome page, and see his page on where to park in Rome. You'd probably have noticed it yourself, but here's the direct link:
http://www.roninrome.com/%20transpor...arking-in-rome
I see there are parking garages in the Prati area - which is great. You'll have an abundance of choices of B&Bs as well as apartment rentals so no need to limit your choices to those that provide parking. When doing your searches, type in Prati, or the Vatican area. Reference the ZTL maps seen in my first link to make sure you're "safe".
I'm aware this doesn't answer the OPs request of finding something near the autostrada - but it would seem to be do-able if he's intent on keeping his rental.