Is it true when flux is at maximum, the coil is moving parallel to the magnetic field?
Magnetic flux is given by

where the angle is measured between the magnetic field line and the surface (not the surface normal here for simplicity).
This is a maximum when the two are perpendicular. AKA if the field is going north-south, you want your coil to be oriented east-west or up-down. If the coil is parallel to the magnetic field...how do you define parallel? If you mean the magnetic field is in the same plane as the coil...there will be no flux, which is what I think you mean.
Put in a different way, if the coil is parallel to the magnetic field, none of the magnetic field goes "through" the coil, if that makes sense. If the magnetic field is perpendicular, all of it goes through the coil.