Had the problem you are having. Four different roofers all checked
roof, said there was no problem.

Was told to:

1) Put a chimney cap on the chimney.

2) Seal the entire brick surface at the end of each summer with a
masonary waterproofing compound (I don't remember the brand I use, it
comes in a gallon jug at the local pro mason store).

3) Light good fires in the chimney during after heavy rainy periods --
since I never used the chimney hardly at all before.

I'm in the Oregon rain forest. I followed that advice in 1995. Haven't
had a problem since.....