On reflection, SAS tells me that Euclidean geometry has a strong semi-local homogeneity, in that every neighborhood of every point is isotropically isomorphic with some neighborhood of any other point --- once you find a good way to say "neighborhood", that is.
The parallel postulate, on the other hand, can be used to construct *canonical* isomorphisms of point(ed)-neighborhoods --- by parallel translation of course; but since the constructed isomorphisms are all parallel in a good sense, we don't get the isotropy structure without SAS. (edit/add:ed): in the other direction, SAS doesn't give any canonical isomorphisms, which is just as well because hyperbolic and elliptical space both have SAS, but not the parallel postulate. (end edit)
The related postulate that Euclid states properly --- that all right angles are equal --- only gives a pointwise isotropy; it doesn't help much for segments subtended by respectively equal segments at equal angles.