But you're describing a well-known intractable problem in computability. Finite-length tape doesn't make the problem any easier to solve, especially within the sorts of time limits that would be acceptable to users of static parsers.
E.g. from Minsky (1967), referring to a machine with a million parts:
"Even if such a machine were to operate at the frequencies of cosmic rays, the aeons of galactic evolution would be as nothing compared to the time of a journey through such a cycle"
So the conclusion stands. If you presuppose an infinite tape, you get equivalence to the halting problem, and if you presuppose a finite tape beyond any non-trivial size, you get complete intractability.
Intractible doesn't mean impossible: perhaps someone will come up with a great new approach. The proof says nothing about what is possible with a finite tape.
I hope you now feel that I have made a point that is, at least vaguely, relevant.
E.g. from Minsky (1967), referring to a machine with a million parts:
"Even if such a machine were to operate at the frequencies of cosmic rays, the aeons of galactic evolution would be as nothing compared to the time of a journey through such a cycle"
So the conclusion stands. If you presuppose an infinite tape, you get equivalence to the halting problem, and if you presuppose a finite tape beyond any non-trivial size, you get complete intractability.