You are selecting the Split To Existing Case option with Case Split, and notice there is no option to select an existing case.
The root cause of this issue is Field-Level Security on the Parent Case field.
To ensure visibility is set for the Parent Case field:
1. Navigate to Setup > Customize > Cases > Fields.
2. Click on the field label for Parent Case.
3. Click the Set Field-Level Security button.
4. Ensure the Visible checkbox is set for the affected profiles and click Save.
5. Reattempt your split to an existing case.