If you use this syntax the user can save the newly added record and stay on the record in edit mode.
var text RID = If([Record ID#] > 0, ToText([Record ID#]), "%%rid%%");
var text URL = URLRoot() & "db/" & Dbid() & "?a=er&dfid=11&rid=" & $RID;
"<a class='SaveBeforeNavigating' data-replaceRid=true style=\"text-decoration:none; background: #1ba802; border-radius: 5px; color: #ffffff; display: inline-block; padding: 0px 0px 0px 0px; width:100px; text-align: center; text-shadow: none; border: 2px solid #1ba802; font-size: 12px \"href='"
& $URL
& "'>Add or Edit Children</a>"
The problem will be that you are saying that the EOTI User has no ability to View or Edit Parent records, only Add. Is there a chance that you can relax that restriction to allow them to have access if the current date time ie now() is within say 10 minutes of the [Date Created]?
The other thought is to somehow leverage the new Feature for Secure links for EOTI https://helpv2.quickbase.com/hc/en-us/articles/29353956195220-Secure-links
So for example, the user enters the Parent name on the form and then you construct a secure link using that Parent Name in the access key formula. Then allow them to edit the record as long as the access key is valid. Does your plan level support Secure Links - I believe you need to be on the Business level plan.