The main problem with only showing emergency contacts is that they are relationships in the system, no different than mother or father. If a person has a relationship already on the record, but it is not marked as an emergency contact; it won't show on the form, so if the parent filling out the form adds this same person as an emergency contact, processing the form will essentially try to add a new record for this person and run the risk of creating duplicates.
Here is a way to get those onto the form
Make sure each emergency contact is in its own Section to keep everything separate. 1. Add a normal relationship field to the form for the purposes of mapping the emergency contact. 2. When dragging over the fields make sure to bring over the Emergency Contact checkbox under Relationship 3. Create a rule based on this checkbox. If the value is No, then hide the section.
This setup will allow you to only display the records that have the emergency contact checkbox marked on the relationship tab of the student record. You may want to also add a couple blank sections at the end to allow the user to add a new relationship as an emergency contact that is not yet on the record.