This question is hard to answer. The first issue is the definition of "quiet" The area inbetween Amsterdam and Brussels is quite densely populated. Yet there are really nice and quite areas even in the cities them self. As @annoyed already mentions, parking is rarely free. In Dutch there is an expression that says that only the sun rises for free, meaning that there is always a catch if something is free. However, both in Amsterdam and Brussels free parking exists. The problem with these places is that you either need local knowledge to find or there is no or limited public transportation in the proximity.
In Brussels there is free parking in the proximity of the atomium and the neighbouring Jette. In Amsterdam you could try finding free parking around Durgerdam.
A better approach would be to find a hotel that offers free parking, or use the P+R Annoyed mentions in the comments. For 8 euro's per day you even get a tram/bus return ticket.
To answer your more generic question: let Google maps be your friend, there are quite a lot of decent options. Examples? Zeist, Kalmthout, Enkhuizen, Hoorn, Lage Vuursche, Lier, Namen, Hasselt, Haarlem, Breukelen (origin of Brooklyn!), Hoboken…