You were right. I tried the same solution on my 2nd server and I still had to copy expat_external.h to /usr/local/include to get a clean make. Looks like the solution is:
1. download, gunzip, tar,
2. run configure,
3. make install (which will fail),
4. move expat_external.h
5. run make install again (which should succeed),
5. install XML::Parser.

And all is well, indeed.

Thanks! You guys are great!