The certificated flight instructor (CFI) was providing primary flight instruction to a student pilot in a tailwheel-equipped airplane under Title 14, CFR Part 91. The CFI reported that during the takeoff roll from a 1,700 foot-long, gravel and turf-covered airstrip, the student pilot aborted the takeoff after he sensed that airplane would not clear a stand of trees at the end of the airstrip. The CFI said that the student pilot closed the engine throttle and applied heavy braking, but the airplane continued into an overrun area that contained a parked airplane. In an attempt to regain rudder authority and avoid colliding with the parked airplane, the student pilot applied full engine power, and the airplane inadvertently became airborne. The airplane veered to the right and the right wing struck a tree. The airplane sustained structural damage to the wings, fuselage, and horizontal stabilizer. The CFI said that there were no preaccident mechanical anomalies with the airplane, and that the decision to abort the takeoff should have been made sooner.