Flight Instructor reported a NMAC while climbing out from an uncontrolled airport on a training flight. The instructor was utilizing a grass runway just parallel to the hard surface runway when another aircraft departed from the hard surface runway and passed the aircraft's wingtip just after takeoff. The instructor performed an evasive maneuver to avoid the other aircraft.

Narrative

At the IPJ airport I was conducting a tailwheel training flight in Aircraft X. I was doing Stop and goes on runway 5 (in the grass beside it) while Aircraft Y was holding short of 5 at bravo. (We had prior permission to take off and land in the grass by the now previous airport manager). We had done two stop and goes (5 second or less stop time) while Aircraft Y was talking to approach gathering the departure clearance. I know this because that was the last radio call he made as we were entering the pattern. On the departure of the second stop and go we started our takeoff roll about 1;500 feet past the beginning of the runway. We were about 40 feet in the air paralleling the runway; just past taxiway Echo and Aircraft Y came flying by on our left side; equal altitude. Our wing-tip to wing-tip clearance couldn't have been more than 40 feet. I took the airplane from my student and broke right flying 10-20 degrees right of runway heading to get some spacing. I radioed the other aircraft in the pattern asking if he made a radio call for his departure; two different people said he did not. We had multiple witnesses to the event including FBO staff and Maintenance on the field. Thankfully I had preached to my student in the ground prior to the flight about how important it was to hold centerline while departing. If either one of us would have [not] held centerline; we could have had a negative aviation event.