As @John mentioned, there is not general way and nice formula for computing the minimum distance of codes. The problem remain difficult for LDPC codes as like as other codes. But, there are several way to construct LDPC codes, as like as finite geometry, design theory, graph theory (prototype graphs), permutation circulant matrices, random construction and other ways. Sometimes if we have more information about the method of construction of LDPC codes, we can say much more. For example, if you have QC-LDPC codes, the girth of the code (shortest cycle in the Tanner graph of code) is at most $12$ and it forces the minimum distance to have nice bound. Also, if you know that your code does not have girth $4$, and it has fixed number of $1$ (say $\lambda$) in each columns of parity check matrix, then the minimum distance is at least $\lambda +1$.
So, it depends what is your way to constructing LDPC codes. Also, the predefined structure can help to write efficient program to find the minimum distance of the codes. Random sampling with MAGMA Algebra is a good way for finding the minimum distance of some structured LDPC codes. Also, MAGMA has function "MinimumDistance" which is useful for finding the minim distance of short length codes.
Anyway, in almost cases we can not find the minimum distance of LDPC codes and we have just some bound. But, because of the special method of decoding of such codes (itterative decoding), there are some other parameters which are important for avaluating the efficiency of codes. Short cycles ditribution, trapping sets, stopping sets, density and some other parameters are very important.