I highly recommend you take the Boston Express Bus! The bus goes from Manchester right to south station. Parking is free at the station in Manchester (last I knew). You can purchase your ticket ahead of time or when you arrive at the station. That’s how I used to commute from NH to Boston for work. Much more comfortable and reliable than using MBTA in my opinion.

Edit: wanted to add there are also stations in Londonderry and Salem if those are easier to get to than Manchester.