Zero article before the word "part" - what is the particular reason for this?

The first thing it say is that you need life and not the life. That’s because you’re referring (I assume) to life in general, rather than to any specific life.

There’s little difference in meaning between part of life and a part of life. Both make generic reference, that is, they refer to part in general, and not to any particular part. The choice between them depends on whether in this case you view part as a singular countable noun, in which case you would say a part of, or whether you view it as an uncountable noun, in which case you would say part of. I would guess that part of was more common.

Both sentences mean similar things but with a slightly different emphasis. The first tends to suggest an important part of the life:-

Being the butt of obscene jokes? It's part of the life [of a pheasant plucker].

Adding the indefinite article adds the connotation that it is a smaller, not quite so important aspect of the life.

Getting tickly feathers down your trousers? It's a part of the life [of a pheasant plucker].