Ok, I've rewritten the trigger so that the logic is separated from the trigger itself.  And on my test class I am getting coverage now, but I keep getting an error relating to an insert statement claiming I don't have a required field set.  I've looked over the objects where this error continues to appear but have not been able to find what is causing an issue.  Also I am not sure if the trigger is written correctly to pass all the newly created objects to the class which does the actual parsing.

Trigger

trigger parseStudentApplication on Student_Application__c (after insert) {

    List<Student_Application__c> studentApps = Trigger.new;

    {

StudentApplicationParser sap = new StudentApplicationParser();

sap.Parse(studentApps);

    } 

}

Class which does the work

public with sharing class StudentApplicationParser {

public void Parse(List<Student_Application__c> sas)

{

List<Contact> contacts = new List<Contact>();

List<npe4__Relationship__c> relationships = new List<npe4__Relationship__c>();

List<Involvement__c> involvements = new List<Involvement__c>();

for(Student_Application__c sa : sas)

{

Contact student = new Contact();

Contact mother = new Contact();

Contact father = new Contact();

npe4__Relationship__c relToMother = new npe4__Relationship__c();

npe4__Relationship__c relToFather = new npe4__Relationship__c();

Involvement__c studentInvolvement = new Involvement__c();

student.FirstName = sa.First_Name__c;

student.LastName = sa.Last_Name__c;

mother.FirstName = sa.Mother_First_Name__c;

mother.LastName = sa.Mother_Last_Name__c;

father.FirstName = sa.Father_First_Name__c;

father.LastName = sa.Father_Last_Name__c;

relToMother.npe4__Contact__c = student.Id;

relToMother.npe4__RelatedContact__c = mother.Id;

relToMother.npe4__Description__c = 'Mother of ' + student.FirstName + ' ' + student.LastName;

relToMother.npe4__Status__c = 'Current';

relToFather.npe4__Contact__c = student.Id;

relToFather.npe4__RelatedContact__c = father.Id;

relToFather.npe4__Description__c = 'Father of ' + student.FirstName + ' ' + student.LastName;

relToFather.npe4__Status__c = 'Current';

studentInvolvement.Contact__c = student.Id;

studentInvolvement.Involvement_Level__c = 'Student';

contacts.add(student);

contacts.add(mother);

contacts.add(father);

relationships.add(relToMother);

relationships.add(relToFather);

involvements.add(studentInvolvement);

}

insert relationships;//errors occur here

insert involvements;

insert contacts;

}

} 

Test Class

@isTest

private class TestParseStudentAppTrigger {

    static testMethod void myUnitTest() {

        Student_Application__c sa = new Student_Application__c();

        sa.First_Name__c = 'StudentFirstName';

        sa.Last_Name__c = 'StudentLastName';

        sa.Father_First_Name__c = 'FatherFirstName';

        sa.Father_Last_Name__c = 'FatherLastName';

        sa.Mother_First_Name__c = 'MotherFirstName';

        sa.Mother_Last_Name__c = 'MotherLastName';

        User u1 = [Select id from User where alias='JTres'];

        System.runAs(u1){

            insert sa;

        }   

        Contact student = [Select c.LastName, c.FirstName From Contact c Where c.FirstName = 'StudentFirstName'];

        Contact father = [Select c.LastName, c.FirstName From Contact c Where c.FirstName = 'FatherFirstName'];

        Contact mother = [Select c.LastName, c.FirstName From Contact c Where c.FirstName = 'MotherFirstName'];

        System.assertNotEquals(student, null);

        System.assertNotEquals(father, null);

        System.assertNotEquals(mother, null);

    }

}