DEPR: parsing to tzlocal
- [x] closes #50791 (Replace xxxx with the GitHub issue number)
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

i think issues have been addressed. Nice thanks @jbrockmendel  I think this needed a rebase? Seeing this on main

```
     def test_parsing_tzlocal_deprecated():
        # GH#50791
        msg = "Pass the 'tz' keyword or call tz_localize after construction instead"
        dtstr = "Jan 15 2004 03:00 EST"
    
        with tm.set_timezone("US/Eastern"):
            with tm.assert_produces_warning(FutureWarning, match=msg):
                res, _ = parse_datetime_string_with_reso(dtstr)
    
            assert isinstance(res.tzinfo, tzlocal)
    
            with tm.assert_produces_warning(FutureWarning, match=msg):
>               res = parsing.parse_datetime_string(dtstr)
E               AttributeError: module 'pandas._libs.tslibs.parsing' has no attribute 'parse_datetime_string'
``` Yah, we'll need to change that to check py_parse_datetime_string have opened a pr