The Eagles intend to move to a new 40,000-seater stadium at the National Sports Centre at their original home in Crystal Palace.
Tottenham's bid to rebuild the Olympic Stadium after 2012, without a running track, includes a proposal to redevelop the athletics arena in south London.
Parish's plans at the moment do not include a running track, but he may have to rethink if Spurs' bid is successful and admits the Eagles' proposed new home could yet become a dual-purpose stadium.
"I don't think we're going head-to-head with Spurs, if they get the nod with their plans it would supercede it I'd imagine," he said. "At that point someone could to talk to us about a greater use for the stadium.
"I haven't seen their plans. Whether their plan represents what is seen as an Olympic legacy and a stadium capable of holding the world athletics championships, I don't know."