I am searching various sources about phylogenetics. I saw some materials about perfect phylogeny and also phylogenies acquired from maximum parsimony constraint. They seem very similar to me. Are they the same?
No. A perfect phylogeny is one such that all characters evolve on the tree with no homoplasy, i.e. for a binary character, changes occur from 0 -> 1 but never from 1 -> 0. A maximum parsimony phylogeny may produce a perfect phylogeny, but typically for real datasets some degree of homoplasy is required to explain character patterns.
Long discussion that took many years to resolve.
MP doesn't account for back mutation, which is a HUGE problem for nucleotide data because in theory 1:4 mutations is a back mutation. Maximum likelihood has resumed is crown here, however Beast deployment of a Bayesian calculation is also very widely, particularly for molecular dating. ML uses a reversible matrix A <-> T, whilst Beast uses a directional matrix A->T and will score T->A separately.
Happy to discuss the approaches and modern interpretations at length.
The model which accounts for back-mutation is the Jukes-Cantor correction (JC correction), this is a basic model present in every phylogenetic algorithm except p-distances). Essentially JC extrapolates back-calculation for the true number of mutations against the observed number of mutations against divergence time. However, when the P-distance (uncorrected observed distance) exceeds 0.75 for nucleotides the model is not viable. Basically, nucleotide divergence >0.75 is saturated and the phylogenetic information is essentially random.