Londondrz,
I happen to believe it is a great idea and have discussed it various people.

Unfortunately all new public footbridges apparently need to be fully accessible (unlike existing ones such as Sydenham Park footbridge), this would mean including lifts in the design. As a result the cost would be massively inflated, just to pick figures out of the air we would probably be looking at £3m for such a bridge, and it is difficult to justify this type of expenditure on a pedestrian bridge.

The only hope would be for a large development above the Perry Vale car park and keeping the public car park underneath, but it seems unlikely. Alternatively we could find a way to explain that the footbridge doesn't disadvantage disabled people, who can have extra reserved bays in Dartmouth Road (but I don't think this works as an argument).

I don't think the Sainsbury's car park often gets full, despite 2 hours free parking. This car park is not far from the shops or the pool, but many people still find it more convenient to drive up and down Dartmouth Road in the hope of a space rather than just parking in Sainsbury's.