What are the consequences of Parity-L = P? 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 &#8484;2 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?
parity-$L$ is in $NC^2$ and parity-$L=P$ would mean that $P$ can be simulated in parallel $\log^2$ time or in $\log^2$ space (since $NC^2$ is in $DSPACE(log^2 n)$)