The computational cost of transfer matrix methods for the Potts model is directly related to the problem of \textit{into how many ways can two adjacent blocks of a lattice be connected}. Answering this question leads to the generation of a combinatorial set of lattice configurations. This set defines the \textit{configuration space} of the problem, and the smaller it is, the faster the transfer matrix method can be. The configuration space of generic transfer matrix methods for strip lattices in the Potts model is in the order of the Catalan numbers, leading to an asymptotic cost of \(O(4^m)\) with \(m\) being the width of the strip. Transfer matrix methods with a smaller configuration space indeed exist but they make assumptions on the temperature, number of spin states, or restrict the topology of the lattice in order to work. In this paper we propose a general and parallel transfer matrix method, based on family trees, that uses a sub-Catalan configuration space of size \(O(3^m)\). The improvement is achieved by grouping the original set of Catalan configurations into a forest of family trees, in such a way that the solution to the problem is now computed by just solving the root node of each family. As a result, the algorithm becomes exponentially faster and highly parallel. An additional advantage is that the final matrix ends up being compressed, not only saving space but also making numerical evaluation on \((q,v)\) faster than in a non-compressed scenario. Experimental results for different sizes of strip lattices show that the \textit{Parallel family trees (PFT)} strategy indeed runs exponentially faster than the \textit{Catalan Parallel Method (CPM)}, specially when dealing with dense transfer matrices. We can confirm that a parallel implementation of the PFT algorithm is highly effective and efficient for large problem sizes...