Parity-L is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero number of acceptance paths), and which is further restricted to work in logarithmic space. Solving a linear system of equations over ℤ2 is a complete problem for Parity-L, and so Parity-L is contained in P.
What other complexity class relations would be known, if Parity-L and P were equal?