The ghost answer on the Internet is scrawled, saying, "When I get the maximum from (2) ..." Look at the topic, where is such a requirement? So, don't look at those answers online! Unless it's scanning!

There should be three F points that meet the requirements. The coordinates of the two F points you calculated are correct (AB is one side of the parallelogram at this time), but you missed one: the vertex D(- 1, -4) of the parabola is also the F point that meets the conditions! ! ! AB is the diagonal of the parallelogram.