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As Alexis King pointed out in comments, it is standard practice to separate parsing from evaluation.

However, to address the current question, it is possible here to evaluate while parsing in an idiomatic way. The key point is the following: lexical scoping without any context-sensitive rules only ever requires a Reader monad, for scope/type checking and evaluation as well. The reason is in the "lexical" property: purely nested scopes have no side effects on other branches of scope structure, hence there should be nothing to be carried around in a state. So it's best to just get rid of the State.

The interesting part is letStarExpr. There, we cannot use many anymore, because it doesn't allow us to handle the newly bound names on each key-value pair. Instead, we can write a custom version of many which uses local to bind a new name on each recursive step. In the code example I just inline this function using fix.

Another note: lift should not be commonly used with mtl; the point of mtl is to eliminate most lifts. The megaparsec exports are already generalized over MonadParsec. Below is a code example with megaparsec 7.0.4, I did the mentioned changes and a few further stylistic ones.