Generally speaking you can fish any river if you're after trout
Check out this link for the rules regarding fishery byelaws in the north-east region (which includes Yorkshire)
Basically you're restricted to fishing worm, in some areas you're prohibitied from fishing a float too but I can't find any reference to that in the Yorkshire bye-laws.
As for where to fish, I'd suggest the upper Swale (around Catterick area) or the upper Wharfe. I think you might find it hard to convince an EA bailiff you're fishing for trout on the lower reaches of the Yorkshire rivers.