Reland "[bookmarks] Clear last used parent cache when adding bookmark fails"

This is a reland of commit eedc2f13b941728513b9dff5d4d9910f94a1c972

Original change's description:
> [bookmarks] Clear last used parent cache when adding bookmark fails
>
> Bug: 415333373
> Change-Id: Ib810fcdb75e190bb8f88411787cd2aa192b06810
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6598838
> Reviewed-by: Brandon Wylie <wylieb@google.com>
> Reviewed-by: Sky Malice <skym@chromium.org>
> Commit-Queue: Evan Luo <evanluo@google.com>
> Cr-Commit-Position: refs/heads/main@{#1467115}

Bug: 415333373
Change-Id: I85fb8fcb9b98776ef536736f96131d64848f5da1
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6605712
Reviewed-by: Sky Malice <skym@chromium.org>
Reviewed-by: Brandon Wylie <wylieb@google.com>
Commit-Queue: Evan Luo <evanluo@google.com>
Reviewed-by: Calder Kitagawa <ckitagawa@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1467240}