I'm not entirely sure you've defined a parabola.

By the conventional definition, a parabola occurs when y is equal to ax^2 + bx + c. You've got y^2=4ax, so the 'squared' is on the vertical axis variable and not the horizontal axis variable. This will give you a parabola lying on its side, and so your definitions of wide and narrow parabolas will also be a bit wonky.

For the parabola

the 'width' you're referring to is governed by a, the coefficient of a. Moving a further from 0 will narrow, or tighten the parabola, and moving a closer to 0 will make it wider, or fatter, until a equals 0 and you've got a REALLY flat parabola (also known as a line).