Parallel (NC) replacements for Gaussian elimination? Suppose one has an matrix $A$ with $c$ columns and $r$ rows with entries in the binary field $GF(2)$. One wants to determine  


its rank, and 
a basis for its null-space.


These can be computed easily in time $O(r^2 c)$ by Gauss elimination, but that is a very sequential algorithm. 

Are there deterministic parallel (NC, polylogarithmic time, polynomial space) algorithms to compute these?

One may be able to assume that either $r$ or $c$ is small (maybe polylogarithmic) if it helps.
Yes, it is in $\mathsf{NC}^2$: 


  Mulmuley, K. A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica 7 (1987), no. 1, 101–104. 


The following (earlier) paper shows that solving a system of linear equations reduces to computing the rank, and thus, together with the above result, solving the system (in particular, finding a basis for the nullspace) is also in $\mathsf{NC}^2$:


  Allan Borodin, Joachim von zur Gathen, John Hopcroft. Fast parallel matrix and GCD computations. Information and Control 52(3):241-25, 1982.