Parallel (NC) replacements for Gaussian elimination?
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.