Custom Post: Support update parent page

[crazytonyli]

Description

Fixes: https://linear.app/a8c/issue/CMM-1957

Testing instructions

Go to a page's Settings, and update the "Parent Page" row.

[dangermattic]

	2 Warnings
⚠️	View files have been modified, but no screenshot or video is included in the pull request. Consider adding some for clarity.
⚠️	This PR is assigned to the milestone 26.8. This milestone is due in less than 4 days. Please make sure to get it merged by then or assign it to a milestone with a later deadline.

Generated by 🚫 Danger

[wpmobilebot]

📲 You can test the changes from this Pull Request in WordPress by scanning the QR code below to install the corresponding build.

	App Name	WordPress
	Configuration	Release-Alpha
	Build Number	31552
	Version	PR #25398
	Bundle ID	org.wordpress.alpha
	Commit	40a3d36
	Installation URL	7g2m0ht51ds0g

Automatticians: You can use our internal self-serve MC tool to give yourself access to those builds if needed.

[wpmobilebot]

📲 You can test the changes from this Pull Request in Jetpack by scanning the QR code below to install the corresponding build.

	App Name	Jetpack
	Configuration	Release-Alpha
	Build Number	31552
	Version	PR #25398
	Bundle ID	com.jetpack.alpha
	Commit	40a3d36
	Installation URL	21a20kch61jn0

Automatticians: You can use our internal self-serve MC tool to give yourself access to those builds if needed.

[wpmobilebot]

🤖 Build Failure Analysis

This build has failures. Claude has analyzed them - check the build annotations for details.

[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud