I got a $50 ticket for parking in front of my house yesterday because it snowed, is there a law in NJ supporting this?

Tells you clearly on the document. Municipal Code 0312 applies.

There are many.  Too many to list all of them here.  Here are two examples:

"Whenever there is a snow accumulation in the Township of Berkeley Heights, no vehicle shall be parked on any public street ..."

"In order that the municipality may have the opportunity to clear the streets and highways of snow and to open said streets and highways in the township to traffic, during the period of thee mergency aforesaid, no vehicle howsoever propelled, shall be allowed to stand or park, whether attended or not..."

I dont believe my road is an emergency snow route.

What you believe doesn't change what the citation states.

For the record, parking restrictions are LOCAL. Merely naming the state is USELESS for determining the rules.

If 2 inches of snow fell or was forecast, you are in violation. 

From the Edgewater Park Township Municipal Code: 

"Aa state of emergency shall exist within the municipal limits of the Township whenever there shall be a snowfall of two inches, or whenever snow is falling and there is a prediction of two or more inches of snowfall. Said emergency shall be presumed to continue for a period of 48 hours after said snowfall ceases."

It's local law, not State law.  CALL AND ASK.

Many many places have a law that you cannot park overnight on the street between certain dates--the typical snow season. It's not necessarily enforced except when it snows, because your parked car interferes with snow removal or plowing.

Where I live, there are signs stating that on every road that enters the town borders, from highways to one-laners.

It's not the town's problem how many cars there are in the household or where you park them, so long as it's legal parking--which yours was not.

You're guilty. Pay the ticket.

Be grateful it wasn't towed!

The public road is not your property