You will first need to create a new household for the parent that will not be staying at the current address. To do this, go to the parent's profile and click "New" to the right of the Household field. Key in the new household name and click "Update Profile". This will now place that parent in a new household and allow you to create a new address and phone number for that individual.
Second, you will want to remove the relationship between the two parents under the "Relationships" section of the profile.
Note: You may need to adjust the student households as well. After the parents are listed in two different households, you can use the "Browse" button to the right of the Household field in the student(s) profile to search for the parent's new household name.