Are there any reasonably priced hotels that offer free parking?

Some near the airport do offer free parking including the Marriott Courtyard South Boston which is reasonably priced IMO.

The Midtown Hotel. It's dated, but we found the room clean and staff very nice. Free parking. T station out front.

Not many, just a few. If you don't mind being outside the city a bit, there are nice hotels in some suburbs that have free parking. Then you can just hope on the T (subway) for an easy ride into the city. I am on the South Shore of Massachusetts so the hotels in Braintree and Quincy pop into my head first, but I know there are others in surrounding suburbs. The ride on the T usually isn't too bad.

To my knowledge, the Midtown (a classic motor inn) is the only place in Boston with free parking. None of the others are "reasonably priced".

But, if you are visiting on a weekend, all garages in the financial district will have a rate for the entire weekend less than one night at the hotel.

As for hotel rates, you must shop like the dickens for "reasonable rates" between April and October.

Try the Best Western Round House suites. It's convenient to Rt 93 but otherwise not a very good location. Long walk to the T but they also have a shuttle.