I do not want a card edge connector that sits on neither the top or bottom of the PCB, I need one that sits parallel to the PCB board.
I think you are after one these "bottom edge, straddle mount" connector types:
We once used some that went to a 5 x 100 row connector on the backplane, had 2 per board. That was a lot of IO! I don't recall what brand it was, that was back in the early '90s.
This connector type is called end launch, and exist in various pitches from many vendors. For connectors with 1.27mm pitch, it is possible to use the standard through hole variant and hand solder them to the board.