In this paper it is investigated which pivots may be processed simultaneously when solving a set of linear equations. It is shown that for dense sets of equations all the pivots must necessarily be processed one at a time; only if the set is sufficiently sparse, some pivots may be processed simultaneously. We present parallel pivoting algorithms for MIMD computers with sufficiently many processors and a common memory. Moreover we present algorithms for MIMD computers with an arbitrary, but fixed number of processors. For both types of computers algorithms embodying an ordering strategy are given.