The fringe analysis studies the distribution of bottom subtrees or fringe of trees under the assumption of random selection of keys, yielding an average case analysis of the fringe of trees. We are interested in the fringe analysis of the synchronized parallel insertion algorithms of Paul, Vishkin, and Wagener~(PVW) on 2-3 trees. This algorithm inserts k keys with k processors into a tree of size n with time O(log n+log k). As the direct analysis of this algorithm is very difficult we tackle this problem by introducing a new family of algorithms, denoted MacroSplit algorithms, and our main theorem proves that two algorithms of this family, denoted MaxMacroSplit and MinMacroSplit, upper and lower bounds the fringe of the PVW algorithm. Published papers deal with the fringe analysis of sequential algorithms and it was an open problem for parallel algorithms on search trees. We extend the fringe analysis to parallel algorithms and we get a rich mathematical structure giving new interpretations even in the sequential case. We prove that the random selection of keys generates a binomial distribution of them between leaves, that the synchronized insertions of keys can be modeled by a Markov chain, and that the coefficients of the transition matrix of the Markov Chain are related with the expected local behavior of our algorithm. Finally, we show that the coefficients of the power expansion of this matrix over (n+1)^{-1} are the binomial transform of the expected local behavior of the algorithm.