We are interested in the fringe analysis of synchronizedparallel insertion algorithms on 2--3 trees, namely the algorithm ofW. Paul, U. Vishkin and H. Wagener~(PVW). This algorithminserts k keys into a tree of size n with parallel time O(logn+log k).Fringe analysis studies the distribution of the bottom subtreesand it is still an open problem for parallel algorithms on searchtrees. To tackle this problem we introduce a new kind of algorithmswhose two extreme cases upper and lower bounds the performanceof the PVW algorithm.We extend the fringe analysis to parallel algorithms and we get a richmathematical structure giving new interpretations even in thesequential case. The process of insertions is modeled by a Markovchain and the coefficients of the transition matrix are related withthe expected local behavior of our algorithm.Finally, we show that this matrix has a powerexpansion over (n+1)^{-1} where the coefficients are the binomialtransform of the expected local behavior. This expansion shows thatthe parallelcase can be approximated by iterating the sequential case.