New haven was a joke. Its 2 hours away.
Your best bet is probably to the off airport parking services. I don't know where you would find cheaper parking nearby, but I guess if you did then you could take the T.
Alternatively, you could park in a neighborhood without resident parking and just pay the ticket. Or I could be wrong and there could be some neighborhoods without resident parking. If so, the "T" goes to the airport for a couple bucks. Silver line or blue line.