Add the Internal ID of the Parent Customer Record to the Custom Record.

1. Navigate to Customization > Lists, Records, & Fields > Record types
2. Edit the custom record with customer record as Parent.
3. Take note of the fieldID of the customer name field.
4. Create a new field.
5. Enter the Name: Customer ID
3. Select the Type as Free Form Text
4. Set Store Value to False
5. Under the Display tab, set Subtab to Main
6. Set the following under the Validation and Defaulting tab:
Default Value = {"ID of the customer name field".ID}
Formula = Yes
8. Click Save

Ex. If the ID of the customer name field is custrecord32 then the Default Value = {custrecord32.ID}