A flight instructor taxiing in a Sling reported the student lost control of the aircraft and the instructor had to take evasive action to avoid another aircraft resulting in a taxiway excursion.

Narrative

While Holding short of Runway XXR number 2 in sequence on [Taxiway] 1 for 5 min waiting for a takeoff clearance a Phenom 300 was cleared to land from a 10-15 mile final; my student set the parking brake and took out their phone to take a picture of the landing aircraft; after the Phenom cleared Runway XXR number 1 was cleared to takeoff; we #2 started rolling before #1 after they repeated the takeoff clearance; student released the parking brake and applied power to move forward accidentally turn the parking brake to the on position; when the parking brake is in the on position it does not allow you to apply brakes in the older model of aircraft; we get within 10ft of the aircraft in front of us the student tries to apply brakes nothing happens we continue to roll I pull power to idle try to apply brakes we continue to roll; I apply full left rudder to avoid the aircraft stopped in front of us; as we start to depart the taxi way I shut down the engine; we come to a stop facing the opposite direction with 2 wheels off the Taxiway 1 on the west side. I check the parking brake now see it set in the on position; set it to off; hold the brakes and restart the aircraft and taxi back to the nearby run up cancel takeoff clearance and requested taxi back to the flight school with Ground; conduct a post flight inspection of the prop and wing verify nothing was hit and there was on damage.The student was a CFI working on their CFII with approximately 20 hours in the type of aircraft; this was my 4th flight with the having completed 2 VFR familiarization flight with the student doing maneuver stalls and landings with the student and was on our second flight conducting IFR training. The student may have not been fully aware of the difference in operation of the parking brake between the older and newer models of the aircraft; I am aware of the differences in parking brakes between did not properly explain the difference between the types of brakes and how they operate; and did not have time to accurately process the information after taking controls to understand that the parking brake was set on and the brakes could not be applied in time before feeling the need to taking corrective action. The proper response to the scenario would be to reduce power to idle verify parking brake is set to off then apply brakes. Why was I unable to do that in the moment a combination of taking control midway through the incident and not having proper access to the parking brake level it was blocked by the student's arm.