No, the fence does not have to be parallel for any operation on the router table.
For the example, lets make a round router table...in this case there is no "Fence parallel to the table edge" (which edge) because the table is round.
You can put the fence ones in one position or turn the fence 30° from this position or turn the fence to 95° from the initial position.
As long as you set the fence 3/8" from the bit and push the wood along the fence, the cut or, slot will be 3/8 from the wood edge (or the fence).
Parallelism is not an issue on router table, it's only important for me, because of the method that I use to set the bit-fence distance.