http://goo.gl/maps/JJLrd
Looks like there is one but it also looks like you have to enter a parks dept area to get to it so I don't know.Might not be a real street.
I know for sure there is a NB entrance a bit further up @ the intersection of Lydig,Bronx Park East and Boston Road.You would go to the end of Bronxdale and then head North on Bronx Park East and left onto Boston Rd and the Parkway entrances directly opposite 2166 Bronx Park East.
http://goo.gl/maps/g22El
http://goo.gl/maps/Up0hZ
Sorry,this above may be only a Southbound entrance, in which case the next entrance is either directly in the Pelham Parkway intersection which you don't want.
Look at this alternative: North on Williamsbridge ( or Eastchester)to Waring,Left on Waring and go all the way to Bronx Park East,left on Bronx Park East for 1 1/2 blocks to an entrance that is just North of Pelham Parkway on Bronx Park East @ Reiss Place.There is never any traffic at all on Waring or Bronx Park East North of Pelham Parkway.
http://goo.gl/maps/mhSFw
http://goo.gl/maps/lvMHM