Since New York doesn't have a residential parking permit system, can a tourist park in Manhattan for free?

In Theory... sure.

In practice...

Unless you have a fairly good understanding of New York's somewhat arcane Alternate Side Parking rules, the patience of a saint, and the willingness to walk all the way across town once you've found a spot - and then to move your car daily and do it all again because of the aforementioned ASP rules - you're not going to find this viable or pleasant.

Another helpful tip, is look for spots in neighborhoods near your destination where street parking is usually more available, then you can hop on a nearby subway line to get to your destination. For example, if you want to go somewhere off the 2/3 or A/C or 4/5 or J/Z line, there is usually free parking between the South Street Seaport and the Brooklyn Bridge - and these spots are a short walk from all these subway lines at the Fulton Street Center (Fulton St. & Broadway). As stated above, just make sure you are not violating the alternate side parking rules on nearby signs, and that you are not blocking fire hydrants or driveways.

A couple other examples: on Riverside Drive from about 72nd Street on the Upper West Side all the way up to Washington Heights - this is a short walk to many 1 and 2/3 train stations. Bonus: there are hardly any driveways.

While the other answers correctly mention that the Alternate Side Parking rules are difficult and arcane, there is no longer the need to actually understand them as a tourist. Instead, you can use one of the online ASP maps to find streets which you can use for free parking at any given moment. For example SpotAngels has the following handy map:

It shows free street parking, paid street parking and parking garages, all in the same view. Most of the free spots will probably be busy by the time you arrive, but it might be worth a shot.

I would like to point out that NYC, Washington D.C. and other "large" meto areas in the New England area are generally considered "public transit friendly."

For example, last time I stayed in D.C. I parked my car at the hotel upon arival, and took combinations of the Subway, Train, Taxi, and Bus anywhere I wanted to go. I didn't drive at all that week.

IIRC NYC and srounding areas are very similar. You typically walk, or take public transit everywhere.

Your hotel should offer you some parking options, though, if your flying in, best to not even get a rental. Just take a taxi instead.