Most popular preconditioners are implicit, that is, to apply them one has to solve a system. One might say that they compute an approximation to the coefficient matrix that is easier to solve with than the matrix itself.
The approximate inverse class of preconditioners is different in that they compute explicitly an approximation to the inverse. Hence the application is an explicit matrix-vector product operation, and therefore trivially parallel.
The method in the SPAI code is based on ideas from :
the minimisation problem
is solved, with the sparsity pattern of M predetermined or adaptively determined. This minimisation problem turns out to reduce to independent subproblems for the rows or columns of M, and is therefore executable in parallel.
An other advantage of this method is that it is not subject to breakdown the way factorisation based methods are.