Random addition method for phylogenetic tree reconstruction
MEGA manual may help to explain the criteria used in selecting the final maximum parsimony tree. 

The random tree additions refers to the generation of random initial trees for a heuristic search to find MP trees. In this case, a tree is generated by randomly selecting a sequence and adding it to the growing tree on a randomly-selected branch.

Instead of using the random tree additions you can use the min-mini algorithm for the initial tree set.