Flight Instructor reported a newly constructed grass strip at BXM is not published in the NOTAMs and presents a safety hazard for parallel operations at the airport.

Narrative

On final with a student to Runway 19L at BXM. The FBO has placed a grass strip between the active runway and the main taxiway; Alpha; which parallels the Runway 19L and 01R. A3 intersects 19L and Alpha and is about midfield. The grass strip is located between A2 and A3 between 19L and Alpha. Landing was full stop and we usually taxi off 19L on to A3 to clear the runway and perform after landing checklist before proceeding. My concern was a grass strip user behind us. The grass strip ends just before A3 and in the event of a botched landing or overshoot; he would broadside my aircraft as it sat on A3.I broadcast my intentions to use A3 as a taxiway while on final. Got no response from the grass strip aircraft so instructed the student to perform a short field landing and was able to turn off A2 before the grass strip guy landed and crossed over to the FBO Ramp to taxi back rather than use Taxiway Alpha to avoid being in close proximity to a landing aircraft. The grass strip was put in place last year unannounced and is not NOTAM'd. It presents a hazard using parallel operations within feet of other aircraft; including solo students.