Which states that parking heading in the direction of the traffic
Even in the one way streets example is states parking in the direction of the traffic. You have to drive on the wrong side or the road to park against traffic. I am not a constitutional law lawyer, but I am pretty sure that illegally driving on the wrong side of the street so you dont see a sign does not grant you 4 amendment protections.
Universal Citation: NJ Rev Stat § 39:4-135 (2013)
39:4-135. Parking; direction and side of street; angle parking; one-way street
The operator of a vehicle shall not stop, stand or park the vehicle in a roadway other than parallel with the edge of the roadway headed in the direction of traffic, on the right-hand side of the road and with the curb side of the vehicle within six inches of the edge of the roadway, except as follows:
a. Upon those streets which have been designated by ordinance and have been marked or signed for angle parking, vehicles shall be parked at the angle to the curb designated and indicated by the ordinance and marks or signs.
b. Upon one-way streets, local authorities may permit parking of vehicles parallel with the left-hand edge of the roadway headed in the direction of traffic, on the left-hand side of the road and with the curb side of the vehicle within six inches of the edge of the roadway.
Amended by L.1951, c. 23, p. 99, s. 75.