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There are two points: the foci.
No there can never be a single point. But yes there are two such points called foci( each called focus) that helps to define an ellipse. An ellipse can then be defined as a curve which is actually the locus of all points in a plane,the sum of whose distances from two fixed points (the foci) is a given(positive)constant . This is further expressed mathematically to obtain the equation of an ellipse.
Just like any other equation, you can set up a table of x values, and calculate the corresponding y values. Then plot the points on the graph. In this case, it helps to have some familiarity with quadratic equations (you can find a discussion in algebra books), and recognize (from the form of the equation) whether your quadratic equation represents a parabola, a circle, an ellipse, or a hyperbola.
I do not think that they have a name. You are talking about two parabola's on top or side of eachother. I don't think they have a specific name but they are not a straight line so you can't call them parallel. I think its just called two parabolas that never cross. Hope this helps!!