  We develop and implement a parallel flatPERM algorithm \cite{G97,PK04} with
mutually interacting parallel flatPERM sequences and use it to sample
self-avoiding walks in 2 and 3 dimensions. Our data show that the parallel
implementation accelerates the convergence of the flatPERM algorithm. Moreover,
increasing the number of interacting flatPERM sequences (rather than running
longer simulations) improves the rate of convergence. This suggests that a more
efficient implementation of flatPERM will be a massively parallel
implementation, rather than long simulations of one, or a few parallel
sequences. We also use the algorithm to estimate the growth constant of the
self-avoiding walk in two and in three dimensions using simulations over 12
parallel sequences. Our best results are \[ \mu_d = \cases{ 2.6381585(1), &
\hbox{if $d=2$}; \cr 4.684039(1), & \hbox{if $d=3$}. } \]
