REF: Reduce duplicative methods between XML parser classes
- [X] Relates to #40131
- [X] [Tests passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests).
- [X] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) _(unnecessary)_
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file _(unnecessary)_

Thanks @ParfaitG 