Finding contacts without relationships (API)
Yes you can! I fiddled with the APIv4 Explorer a bit and came up with this:

The trick is to join onto RelationshipCache with the relationship type(s) you are wanting to exclude (note that the double quotes around '&quot;Parent of&quot;' are needed because you are passing a string literal not the name of a field or other SQL expression). And then in the WHERE clause specify that you want the relationship_cache.id to be NULL.
The GroupBy is necessary because your join can potentially produce duplicate results if the contact has more than one relationship, although technically that's impossible with this query so, meh, you could try leaving it off.