Parallel is parallel. If the two mirrors are parallel then the light will bounce between them and stay at the same level.
I agree with this - that's why the experiment will be inconclusive.
BUT. If there is some upward force acting on light then you'd expect the light to rise even if the mirrors are parallel. If there isn't then you wouldn't.
Given your previous statement on how we establish that something is parallel, this is necessarily false. If you want to present an alternative definition of "parallel" (note that it cannot refer to optics or lines perpendicular to the Earth's surface), and if you can propose a setup in which this can be achieved, I might be interested.
Do you have some other definition for parallel mirrors beyond "Two mirrors who are equidistant from each other at all points"? Or alternatively "A set of mirrors whereupon any two long edges are equidistant from each other at all points along the edge"? The first assumes two mirrors facing one another, the second assumes mirrors arranged in a circle for some reason and oriented such that, with a mirror in the shape of a rectangle, the short edges point 'down' and 'up' but their orientation is only relevant in regards to other mirrors in the circle. Meaning 'down' and 'up' do not need to refer to the direction of the Earth's surface.
As I already explained in the post above. If the EA is true, the mirrors in a resonator that keep the light inside the resonator would not be physically parallel, they would be slightly tilted with respect to each other to compensate for the EA effect.
But if you have aligned this resonator once and you rotate now the whole setup let's say by 90° around it's optical axis, you would have to realign the two mirrors in order keep the light still inside the resonator. But that is not the case if e.g. anyone can observe who takes the above described HeNe laser or any equivalent resonator device.
But you can also think of other experiments. E.g. diffraction experiments with light or x-rays. If the target is symmetric in the horizontal and vertical direction, the diffraction pattern will have the same symmetry. But if EA is valid, the diffraction angles in the vertical direction will change with distance to the detector, the pattern will become asymmetric. Also something no one has ever observed.
Or take a laser beam with a slightly divergent nicely round TEM_00 mode and let it propagate over a long distance. The parts of the beam with different vertical divergence angles will be slightly differently affected by EA, but no the horizontal components. So the beam will be distorted from the initial round shape. Also never observed.
Or take the resonators in the two arms of the LIGO interferometer. The mirrors are 4km appart, if they would be tilted to compensate for the EA effect, you would directly see the tilt of the mirrors. Once heard a detailed talk from one of the people involved in the mounting of the mirrors. I'm pretty sure he would have noticed that.
I'm mean you have some many applications where it is extremely crucial to be sure about your alignment of your light, x-ray, Thz or microwave, etc. beams down to atomic length scales, and no one has ever noticed an asymmetry in the vertical direction on propagation of the beams...