A training flight in the LBB traffic pattern for Runway 8 initiated a go-around after observing traffic on Taxiway Lima in a protected approach area to the landing runway.

Narrative

Training flight in Aircraft X in LBB traffic pattern (touch and go's). On right downwind to Runway 8; we were cleared for the option to Runway 8 and advised that there was Aircraft Y on a 1 mile final to Runway 17R. I had the traffic in sight and Tower advised us that the traffic was no longer a factor and we were still cleared for the option to Runway 8. We continued the approach and on about a 1/4 mile final to Runway 8; my student said 'they're still in the approach path.' I saw that Aircraft Y was taxiing on Taxiway Lima and was still in the protected approach path area to Runway 8 that is marked on Lima. I immediately initiated a go-around to avoid flying low over Aircraft Y. The closest that we got to Aircraft Y was approximately 125 feet laterally and 200-250 FT vertically. After entering right traffic again; I told the tower controller that we went around due to Aircraft Y being in the approach path and asked him what happened. He said that the ground controller had not told Aircraft Y to hold short of the approach path to Runway 8. He did not say that I had done anything wrong (we were never told to go-around or that our option clearance was canceled during this situation). I have called the Tower before to discuss the approach path and when you have to hold short of it. I was told that anytime an aircraft was overflying Lima (taking off of Runway 26; or landing on Runway 8); that aircraft on Lima would have to hold short of the approach path to 8. I was also told that aircraft would NOT have to hold short of the approach path to 8 if an aircraft was landing on Runway 26; or taking off of Runway 8. Knowing this; I was especially confused why Aircraft Y had gone into the approach path with our aircraft on short final to Runway 8. I think that the contributing factors were that this is a training facility for both pilots and controllers; as well as Aircraft Y possibly not checking the final approach path before proceeding through the protected area. Based on the explanation I received a few weeks back; the Controller made a mistake in this situation by not having Aircraft Y hold short of the approach path. I also believe that I could have detected the possible incursion earlier had I not been so focused on my student's glide path and airspeed control. Even though we were not in any immediate danger of hitting the other aircraft; it is always nice to initiate a go-around as early as possible for something like this.