What is the Beggs/Mueller emergency spin recovery technique?
The Beggs/Mueller technique is:


engine idle
let go of the stick
push rudder against spin direction


This was developed for pilots of the Pitts Special biplane and will work well with naturally stable, powered aircraft. With gliders (where it strictly doesn't apply) it will be less effective, but still work in most cases. The better way is to combine both, rudder against the spin and stick forward (your PARE technique), but only until the rotation stops. Then both the stick and the rudder must be neutral until speed is sufficient for the pull-out. Beggs/Mueller is more about avoiding wrong inputs due to confusion than ending the spin in the most direct way.

Spin recovery with the hands off the stick will be slower than with active control input. If the control surfaces are free-floating, natural stability is normally less than what it is with fixed control surfaces. For that reason, the Beggs/Mueller technique is less effective than actively ending the spin with both rudder and elevator. If the pilot does not realize he is in an inverted spin, pushing down will make matters worse, however.

Why rudder against spin direction?

An aircraft spins because something stabilizes both the rotation and the high pitch angle. First the rotation: This is caused by fully separated airflow on the retreating wing while some section on the advancing wing has attached airflow. The high drag on the retreating wing pulls this side of the aircraft backwards, while the lift on the section of wing with attached airflow pulls that side of the aircraft forward. Remember, airflow is mostly coming from below, and the spin motion modifies the local flow direction the more one moves away from the center of the aircraft.

Why elevator forward (Back in inverted spin)?

Now the pitch angle: The center of rotation is ahead of the wing in the forward fuselage. The masses in the forward fuselage produce little centrifugal force, while the masses in the rear fuselage and the tail, having a substantial distance from the center of rotation, produce a substantial centrifugal force which pulls the whole aircraft into a level attitude. Since the aircraft's motion is mainly downward, this inertial moment increases the angle of attack above the stall angle.

To break the spin one needs to end the rotation or the pitch attitude. Since they mutually reinforce each other, one must be ended to end both. On aircraft with long fuselages the rudder is more effective to stop the rotation, while on aircraft with long wings the elevator will be more effective in ending the spin via reducing pitch attitude. The rotation is then stopped by the roll damping of the wing. Using both together will give the best effect, however.

Why engine to idle?

Having some added dynamic pressure on the vertical tail will also help to end the rotation, but then the aircraft will pick up speed very quickly. Setting the engine to idle will avoid falling into this particular trap, but again for the price of a somewhat slower recovery. Sometimes an aircraft needs that added dynamic pressure to end the rotation, so Beggs/Mueller will not work in all cases.