A pair of parametrized surface are isometric if and only if all corresponding curves have same length.
I have proved one direction, unable to prove the direction a pair of parametrized surface are isometric if all corresponding curves have same length.
Presumably you have a diffeomorphism between the two spaces (otherwise what does corresponding mean).
It would be true if the distance from $p $ to $q $ is the infimum of the distances of all curves from $p $ to $q $... but I believe this is true. ..