Firstly the "Grandfather paradox" is not "flawed" it is a set of mutually inconsistent assumptions. Given they are not mutually consistent then one of the assumptions must be false. Which is a matter of opinion.
But this paradox is not the motivation behind parallel universes theories. In so far as parallel universes are taken seriously at all they are invoked by certain
interpretations of quantum mechanics to preserve certain other desired assumptions which the interpreter wants to keep.
You can't "prove" anything until you have a working time machine to test. You can at best "prove" what a given theory predicts. Within what theory are you working? Classical Einsteinian GR?
The paradox is "solved" by invalidating anyone of the assumptions used in its formulation. The most obvious is "cosmic censorship" i.e. time travel is impossible.
If you like I'll give you my "resolution". Consider say water freezing around a sphere of ice. Think of the patterns of defects in the ice as the sphere grows as "objects". There is a certain randomness to their formation but they also propagate according to some definable rules as the ice forms. Think of these rules as "dynamics" think of the ice as "the past" and think of the liquid as "the future". Now building a time machine is simply a matter of going back and melting down to a point inside the ice. Doing so "erases" all the future after this event point. The ice will refreeze consistent with the "rules" but possibly with a few artifacts from the act of resetting and using the leeway given by the randomness.
The time traveler will have whatever memory he carried with him but his history will be rewritten including his appearance in his recalled past. He must integrate himself back in the time line consistently with the laws of nature (i.e. he appearance must still conserve energy, charge, momentum etc.) so his appearance must look like a very very unlikely random accident (allowed by quantum uncertainty).
I am of course invoking a form of "meta causality" and "meta-time" in which the dynamics of the water and ice and freezing process is defined along with the dynamic physical time defined by the radius of the sphere of water and propagation of "material" defects. One then must ask the meta-question of whether a meta-time machine is possible and a meta-grandfather paradox needs resolution.
So really it doesn't resolve anything except, as with Everett's many worlds interpretation of QM, providing the speculator with a meaningless world picture into which he can integrate the assumptions he wants instead of sticking to operationally meaningful science.