One approach is to ask around for a neighbor with infrequent use of car late at night and arrange ongoing off-street parking, but commit to moving on-street by 7:00 AM or so. Landlord may offer ideas. Meet occasional short-term needs by arranging pull-in to a driveway of co-operative neighbor from 11:00 PM - 7:00 AM on selected nights to enable plowing overnight. In any case, w/need to move by 11:00 PM but restrictions stop at 7:00 AM. Final authority is town website http://www.montague-.../MontagueMA_DPW/snow, or Public Works phone 863-2054, but I believe default is parking be off-street. DPW is reasonable, but Police and Public Works have final say.
BTW, welcome to Montague and you have chosen a good town to live in. The Crossroads of the World is Ave A and Third St., but close behind are Bridge & Rte 63 in Millers Falls and Main & North St in Montague Center.