We parked at the Post Office Square Parking Garage because it was a short walk away but the cheapest hourly parking rate we could find close to the aquarium. Don't remember the price we paid, sorry.
There is also a parking garage withi the block. I believe it was $40
Park at any lot most are hourly so check when you park the garage for the aquarium is about 36 for the day
It's expensive but the parking garage is right next to the building
Don't forget about tolls. If you are not Boston resident you don't have device which automatically pay tolls in the city
There is a garage right at the aquarium 40 dollars
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