BUG: Fix index bug due to parse_time_string GH24091
- [x] closes #24091
- [x] tests added / passed
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [x] whatsnew entry

In Issue #24091 I detailed the root cause of the bug, and implemented the @toobaz suggestion.

I added a single line to test this case in the existing test, ran the full UT suite, checked the linting, and added a whatsnew (I added the whatsnew in the other category, as It wasn't clear where it belonged).



Hello @SimonAlecks! Thanks for submitting the PR.

 - There are no PEP8 issues in the file [`pandas/tests/tslibs/test_parsing.py`](https://github.com/SimonAlecks/pandas/blob/518545d06f5f2c7fccadea4ebf218b5a904bcd31/pandas/tests/tslibs/test_parsing.py) !

 # [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=h1) Report
> Merging [#24294](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/7b0fa8e94e22aa4a491fa574bc56f8ca791f7608?src=pr&el=desc) will **decrease** coverage by `<.01%`.
> The diff coverage is `n/a`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/24294/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #24294      +/-   ##
==========================================
- Coverage   92.22%   92.22%   -0.01%     
==========================================
  Files         162      162              
  Lines       51824    51824              
==========================================
- Hits        47795    47794       -1     
- Misses       4029     4030       +1
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.62% <ø> (-0.01%)` | :arrow_down: |
| #single | `43.01% <ø> (ø)` | :arrow_up: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/indexes/datetimes.py](https://codecov.io/gh/pandas-dev/pandas/pull/24294/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9kYXRldGltZXMucHk=) | `96.17% <0%> (-0.16%)` | :arrow_down: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=footer). Last update [7b0fa8e...518545d](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 # [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=h1) Report
> Merging [#24294](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/7b0fa8e94e22aa4a491fa574bc56f8ca791f7608?src=pr&el=desc) will **decrease** coverage by `<.01%`.
> The diff coverage is `n/a`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/24294/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #24294      +/-   ##
==========================================
- Coverage   92.22%   92.22%   -0.01%     
==========================================
  Files         162      162              
  Lines       51824    51824              
==========================================
- Hits        47795    47794       -1     
- Misses       4029     4030       +1
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.62% <ø> (-0.01%)` | :arrow_down: |
| #single | `43.01% <ø> (ø)` | :arrow_up: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/indexes/datetimes.py](https://codecov.io/gh/pandas-dev/pandas/pull/24294/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9kYXRldGltZXMucHk=) | `96.17% <0%> (-0.16%)` | :arrow_down: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=footer). Last update [7b0fa8e...518545d](https://codecov.io/gh/pandas-dev/pandas/pull/24294?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 can you merge master and update Closing as stale. Ping if you'd like to continue