(1) Because m > 0, so -m < 0, so the intersection of parabola and Y axis is on the negative semi-axis, and there is an analytical formula that the symmetry axis x=2, so C(0, -m), C'(4, -m).

(2) Because the parabolic opening is downward, P and Q cannot be below C and C'

I) When P is on the right side of the symmetry axis, according to the parallelogram property, CC'=PQ=4. Because Q is on the axis of symmetry, the abscissa of Q is 2, so the abscissa of P is 4+2=6. Because P is on a parabola, if you bring 6 into the analytical formula, you can get that the ordinate of P is 24-m, because CC'‖x axis, CC'.

II) Similarly, when P is on the left side of the symmetry axis, the coordinates of P (-4,24-m) and Q point are unchanged.

(3) Let the intersection of CC' and the symmetry axis be m, so CM=2. Because QM=24-m, CQ can be obtained by Pythagorean theorem, and because CC'=4, the perimeter of parallelogram can be obtained.

I don't know if the calculation is correct, the idea is basically like this ~