At this time, invoices can only show the Primary Parent/Parent 1, and this is hard-coded and cannot be changed. 

If this is sensitive for your families, a workaround is necessary:
  1. Navigate to the family record. 
  2. Pull out the blue "What Can I Do?" menu from the left.
  3. Select More Options.
  4. Click Update Family And Student Information.
  5. Enter the full name of Parent 1 into the Parent 1 First Name field.
  6. Enter the full name of Parent 2 in the Parent 1 Last Name field. 
  7. Click Save Changes.
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This is then how the names on the invoice will generate.

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Keep in mind this will impact how you search for the family, if by name (ex. from Families > Manage Families) as the Parent 2 name will be located in the Last Name field.