BUG: Parent taxonomy term auto sync bug

[alaa-alshamy]

Describe the bug

When linking a child taxonomy term with a post, ElasticPress will add the parent and child terms info to the post document in Elasticsearch, but when edit the info of the parent taxonomy term for example changing its slug it will not be synced automatically to the post document in Elasticsearch unlike when editing the child taxonomy term slug it will synced automatically.

Because of that if I queried posts via the new parent taxonomy term slug it will not work until I make a full manual sync via Sync feature of ElasticPress because it will fix the issue via update the parent term info.

Steps to Reproduce

1. Go to WordPress default categories
2. Add 2 terms, one of them child to the other
3. Go to WordPress default posts
4. Add a new post and link the child term with it
5. Edit the parent term slug
6. Go to Elasticsearch and check the post document and you will find the old parent term slug is still exist
7. Optional: Make a full sync via sync feature of ElasticPress and check the post document again and everything will be ok

Expected behavior

The parent taxonomy term info should be synced automatically when it's got changed

Screenshots

Environment information

• Device: Laptop
• OS: Ubuntu 22.04
• Browser and version: Chrome 107.0.5304.87
• WordPress version: 6.0.1

• ElasticPress version: 4.3.1
• Elasticsearch version: 7.10.2
• Where do you host your Elasticsearch server? Local docker setup
• What ElasticPress features do you have currently active? Post Search, Custom Search Results, Related Posts, Protected Content, Terms
• Plugins and version:
• Theme and version:

• Site Health Info:

Additional context