What is the complexity of the following problem?
As Ariel explained in the comments:

"then $a$ indeed represents a valid computation of $M$ on $w$ since the Cook Levin reduction is parsimonious" - This has nothing to do with the reduction being parsimonious, this is a result of the specific construction of the reduction. Also, you don't have to appeal to Cook-Levin to encode computations of a Turing machine, there is no real reason to talk about assignments for boolean formulas in this context. You can simply say that $a$ is a description of $M$'s tape, state, and head location during some prefix of its computation.

Additionally, the input should consist of $M,w$ alone, you don't have any information about $M$'s running time from its encoding (how would you know if the machine actually runs in exponential time?). Now, the problem is trivially solvable in polynomial time by reading $a$ and checking that any configuration is reachable from its predecessor by one of $M$'s transition functions.