Allow role to add a new record but not modify existing records on a connected table

Have you tried Role Permissions? You can give the user the ability to add records but not to edit records.

I have looked at permissions.  My sticking point is that I only want to restrict one filed and not lock them out from modify all fields.  I haven't been able to determine how to create custom access to only lock it out from modifying the field.  The only custom option for a field is view or modify.  

Have you tried a form rule that says when the user is in Role XYZ and when this certain field is not Blank, , make read-only that field.

The rule I tried on the child table is when the user role is equal to XYZ and create date for the child record is on or before now (So anything that's not new) make the field read only.  

The date created will be blank until the record is saved. So you need change that to where the [Date Created] is not blank, make read only.

I did try that as well and it didn't work.  I have also tried other logic where a field is completed after the initial entry and it still doesn't fix the issue.  The issue seems to be that it's not carrying the rule from the child table that is being displayed on the parent table.  The user can edit the related child record from the parent table, but they can't edit the record if they are on the child table. 

Ate you referring to Grid a Edit editing of the children?

Yes.  I have the table on the Parent record
 

I think in the end, for this particular problem, you will need to choose between allowing Grid Edit and controlling editing on that single field. So one solution is to disable Grid Edit on that form and then when your Users go to edit they will have to Edit on the form where the Dynamic form rules will apply.

Dynamic form rules are called dynamic form rules because they only apply dynamically on forms, and not in Grid Edit mode

Thanks for the help