Skyhawk 172/Cutlass 172
Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Commercial; Flight Crew Multiengine
Flight Crew Last 90 Days 150; Flight Crew Total 560; Flight Crew Type 555
Fatigue; Distraction; Situational Awareness; Training / Qualification; Physiological - Other
Conflict NMAC; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action; Flight Crew Became Reoriented
Airspace Structure; Human Factors; Procedure
My student and I were inbound to ZZZ during her training so she was under foggles. We asked approach for the Localizer approach to the left runway but they said unable due to multiple arrivals and told us to remain visual. We loaded the approach anyways for guidance and I told her if we could fly over the FAF then she could log it. Tower put us on a heading vector for left base which had us set up 90 degrees from the FAF with the nose pointed barely north of it. We were cleared to land and told not to overfly final. Autopilot was on and I told my student to leave it in heading mode so we could control our intercept angle. I made visual contact with the jet airliner on final for the right runway; they were to the right of us while we were still on base. They were close but not to the degree that it alarmed me. I think my sight pictured matched what I assumed was about to be a turn to final. With my eyes outside; I didn't realize my student was waiting for the localizer to come alive to start the turn toward the runway; so when it did start to move we blew through it quickly due to the 90 degree intercept angle. Since we were making that turn with autopilot on we were turning at standard rate and about two seconds later were dangerously close to the final for the right runway. I lost visual on the jet as we started turning. All at the same time Tower started yelling and my student cut autopilot off and put us in a steep bank to get us back on final for the left runway. I think she saw the jet through her foggles out her window on the right. We heard the jet go missed; I think he said something like 'they were just a little too close for comfort.' I told her to go visual and she landed.I learned a lot from this. I started to feel sick earlier today and my energy level deteriorated rather suddenly before we flew. I'm dealing with an allergy cold and have felt foggy and disoriented since leaving work. It was stupid to try to fly over a FAF for the sake of logging an approach. I could have paid better attention to what was going on inside the cockpit rather than fixating on the jet once I had identified it's location. Sickness caused processing in my head to slow. I am certain I would not have flown if I was going for a joy flight solo; but having work to do and a students' checkride to prepare for was a massive external pressure that I could not brush off. I believe having autopilot on made my student subconsciously complacent and forgetful that we had not activated the NAV and APR modes on the G1000. And I don't believe traffic alerts work quite the same in the G1000 when an approach is activated but I'm not sure. I'm usually overly careful with parallel approaches as I'm based at this airport and have been in this situation many times. Somehow today my reaction time wasn't what it normally is. I'm grateful the pilots of the airliner had been paying attention as we could have caused something more serious.
[Report narrative contained no additional information.]
C172 pilots reported flying through the final approach course and into conflict with an air carrier on the parallel runway.