Time Reparametrization Invariance in GR

"Symmetries" in the name gauge symmetries is a misnomer. A better name is gauge redundancy. Whenever the fields in our system are redundant, i.e. distinct field configurations correspond to the same physical configuration, we say that the system has a gauge symmetry. Arguably the most famous example of gauge symmetries are those found in the standard model. In modern terms, the physical configuration of these systems is describe by some abstract connection on a principal bundle (see lectures 21 and 22 of this playlist), which is a 1-form on the bundle. Gauge redundancies appear due to us trying to describe this connection as a field theory on spacetime, which involves replacing the 1-form on the bundle to a 1-form in spacetime $A_mu$. In order to do this one needs to choose a local section of the bundle, which is the redundancy in the description. Changing the section chosen changes the 1-form in spacetime by $A_mumapsto A_mu+partial_muLambda$ (in an Abelian theory).

However, there are many other types of gauge theories. An example is a relativistic particle in spacetime. Conceptually, the simplest form of this theory is that of a 1 dimensional submanifold of spacetime (depending on the theory one is consider one might require this submanifold to be timelike, etc..), describing the trajectory of the particle. At this stage this looks very different from a field theory, since the basic objects are submanifold. In order to put it in field theoretical terms, what we do is that we invent the abstract notion of a worldline, which is some 1 dimensional manifold whose existence is independent of spacetime. Then, the theory can instead be understood as one of embeddings of this manifold into spacetime, where the image of such an embedding is the trajectory the particle follows. The embeddings can now be understood as fields (which are however defined on the worldline, not on spacetime), and thus we have succeeded in describing the particle as a field theory. However, in doing so we have introduced a redundancy, namely the choice of a wordline. Since this worldline has no physical significance, the theory should be invariant under changing the chosen worldline by a diffeomorphic one. This is a point of view towards reparametrization invariance which makes it clear that it is a redundancy and, therefore, a gauge symmetry.