Can a parabola have 2 points on y axis instead of on the x axis? would it even count as a parabola?

yeah, a horizontal parabola can intersect the y-axis at two points...

for example,

 x = y²-4

intersects the y-axis at (0,2) and (0,-2)...

it is a parabola but not a function...

A parabola Could be presented in cartessian plane by an equation in two forms

1) function form:

f(x)=ax^2+bx+c (a#0)

one value x --->one and only one value y

--->one y-intercept

2) line form: y^2=2p(x+h)

---->one value x --->0,1,2 value y

---->0,1,2 y-intecept

Yes, a horizontal parabola can intersect the y-axis at two points.

For example,

 x = y²-4

intersects the y-axis at (0,2) and (0,-2).

It is a parabola but not a function.

No, because that wouldn't be a function, since it doesn't pass the "vertical line test"

Yes, you can have a horizontal parabola, but it wouldn't be a function.

Example:

x = y² - 1

It can not. A parabola is a function, and in order for a function be to be a function, it must pass the vertical line test. In other words, a function can only have one output, y, for each unique input, x. I can't have (2,4) and (2,-3) both be existing coordinates on a function. Because I am limited to one y for each x, I can only at most have 1 y-intercept for any given parabola.

However, there are constructs called inverse parabolas which describe parabolas that are basically turned sideways, but these are typical parabolas per se because they aren't functions.