The only mainstream GPS unit I ever had that had any ability to handle this was my Garmin StreetPilot c550. You could tell it you were driving a truck and it would avoid parkways in NY. It would also avoid the Garden State Parkway in NJ, which was inappropriate in my case. Incidentally, Garmin support indicates that the truck setting is supposed to do exactly the opposite of this, and actually prefer "highways", but in NY it seems to work as I described.
Part of the problem is that vehicle definitions and roadway restrictions vary by jurisdiction. For example, in NYC a truck is defined as a vehicle having more than two axles or more than 4 wheels, and commercial vehicles are a different classification; but in NJ a truck is typically defined as a vehicle with a registered weight exceeding 3.5 tons.
I spent several hours one day (quite a while ago) using Map Share to edit my maps to prevent routing on Parkways in Staten Island, Brooklyn and Queens. I basically blocked travel in both directions on these roads. It's tedious work, since you have to do this for every road segment - there is no way to select the entire road at once for this type of change. Also, if you're doing this for more than one device, you have to do it on each device separately, since the maps are keyed to each device.
I just found out earlier today that ALK has a standalone navigation device in their PC Miler product line, which incorporates roadway restrictions for commercial vehicles. Device pricing seems comparable to TomTom, but I have no idea about the pricing for updates of live services. I looked at their products a few years ago when it was software only (they required a PDA or smart phone), and they told me that for my needs (a commercial vehicle registered at under 3.5 tons), their truck-specific products were overkill. You might want to give them a call and see how their current products can be applied for your situation.
I rarely have to travel into NYC, so the restrictions you speak of don't affect me much, but I do sympathize, as I have had to deal with this myself.