  We present, as a proof of concept, a way to parallelize the Clifford product
in CL_{p,q} for a diagonalized quadratic form as a new procedure `cmulWpar' in
the \Clifford package for \Maple(R). The procedure uses a new `Threads' module
available under Maple 15 (and later) and a new \Clifford procedure `cmulW'
which computes the Clifford product of any two Grassmann monomials in \CL_{p,q}
with a help of Walsh functions. We benchmark `cmulWpar' and compare it to two
other procedures `cmulNUM' and `cmulRS' from \Clifford. We comment on how to
improve `cmulWpar' by taking advantage of multi-core processors and
multithreading available in modern processors.
