That is the definition of a function, but it's purely a formalism. In reality, and in theoretical math, we are frequently interested in functions that generate multiple values, and it's very easy to represent those in terms of the formal definition:

1. When we want to consider both positive and negative square roots, we can just say that instead of the function sqrt(x): ℝ ⟶ ℝ which always gives a nonnegative real number, we are using a function full_sqrt(x): ℝ ⟶ ℝ² which gives an ordered pair of real numbers.

2. We mostly define arcsine as yielding a single value between -pi/2 and pi/2, but it's just as easy to define the equivalence relation "x ~ y when sin(x) = sin(y)" and then say that the unique value given by the arcsine function is one of the equivalence classes which that relation induces over the real numbers -- or that it is the subset of the reals included in such an equivalence class.

3. If we are concerned about graphing a multi-valued function, we can do that too; for your parabola, we can define the three-dimensional function "z(x,y): ℝ² ⟶ {0, 1} = 1 when x = y² and 0 otherwise" and the graph of this function is the parabola you want.

So you're wrong about the definition of a function, except that you're right in spirit; the way we normally think about functions, as opposed to the way we define them, definitely allows them to yield multiple values for one input.

You're catastrophically wrong about bijective functions; those are an important concept and your definition is not related to the actual one. A bijective function is one for which all values in the target set are reached by exactly one, no more and no less, value in the source set.