How can I search for notes using the built-in search plugin, which supports regex, to search for notes that have no entry in specific keys within the frontmatter?

I am using templater and thus am using my frontmatter looks like this:

---

aliases:

publish: false

tag:

---

%%

date:: [[2023-07-11 Tue]]

parent:: [[XYZ]]

%%

Searching for "parent:: " works fine, but I want to find notes that do not have a parent note yet. For this I tried parent:: /$ as indicated here and here, but this and other variations like parent:: /\$/ do not yield any results.

Where am i going wrong?