From wiki: "In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle".
I arbitrarily took 4 points [cos(-Pi/6),sin(-Pi/6)], [cos(Pi/4),sin(Pi/4)], [cos(3*Pi/4),sin(3*Pi/4)], [cos(-Pi/2),sin(-Pi/2)] on the unit circle. A total of 4 solutions were found, in two of which the parabolas F and G coincide. In the figure - the first solution from the list L1:
Parabols:=plots:-implicitplot([F,G], x=-3/2..2, y=-5/2..3/2, color=[red,blue], gridrefine=3):
Points:=plots:-pointplot(P, color="Red", symbol=solidcircle, symbolsize=14):
plots:-display(Parabols,Points,Circle, scaling=constrained, size=[500,500]);
Addition. In fact, we have the only solution. In solutions L and L , parabolas F and G differ only in the order of succession.