SR20 instructor pilot with student reported an airborne conflict while attempting to land at a non-towered airport. The instructor states they were cut off from their Runway by another aircraft practicing engine out landings on a parallel Runway. The other aircraft denied any conflict when questioned over the radio.

Narrative

My student and I were doing pattern in ZZZ on Runway XX. Another airplane from ZZZ1 was also doing pattern. They were doing landings on the grass next to Runway XX and making radio calls for 'Runway XX grass'. My student and I departed Runway XX and within 10 seconds after our departure; the other plane was already departing from the grass next to Runway XX. Throughout the entire pattern we were ahead of the traffic but he was VERY close to our tail following behind us; with very little spacing. Our traffic alerting system was going off for the entire downwind; base and final. We made all radio calls for each leg of the pattern. Upon making our call 'turning final Runway XX'; the other plane called on the radio and said 'Aircraft X turning final Runway XX grass; we are simulated engine failure and will be cutting in and landing in front of you'. At this point my Student and I were 1/2 - 3/4 mile from the touchdown zone and were forced to do a go-around due to the other plane cutting us off. I made the radio call 'Aircraft Y going around Runway XX; and we would appreciate if you didn't cut us off on the next trip in the pattern'. The other aircraft responded with 'We aren't using the same Runway as you'; to which I did not respond. When we made the decision to go-around we were within 500 ft. of the other aircraft and would have been within 100 feet if we continued to land parallel to them. They were not using the Runway but the grass next to it; which does not allow sufficient spacing for a parallel approach. We were 1/2 mile ahead during the entire pattern and the other aircraft did not use proper right-of-way procedures resulting in a NMAC on short final.