Hi DeuceJ! I'm a little tired, so forgive me if I make a mistake, but I think I can offer some help.
Finding the area of this region will involve some integration. To find equations for the parabolas, start with , where is the vertex and is the distance between the vertex and the focus or directrix; for the horizontal parabolas just interchange and .
Now, since the region is symmetric as you seem to have observed, consider one corner. The two parabolas that lie in that quadrant can be written as a function of , as long as you only worry about one half of the horizontal one. Find their intersection, and integrate along the appropriate intervals. Then remember to multiply by 4 to get the total area.