You are showing the light source at the apex of the parabola. It should be at the focus. E.g. if you take a line from the lamp at right angles to the parabola's axis, it should strike the parabola at 45 degrees. Maybe it's just the way you have drawn it.
According to my algebra (messy, so unreliable), of the light that is reflected, the intensity at radius y from the centre of the target area varies as ##\frac{4f^2}{(y^2+4f^2)^2}##, where f is the focal length. To that you need to add the light that arrives directly.
So as suspected, the intensity diminishes as you move away from the centre of the target area.
Consider a small area radius y at the centre of the target, the target being distance x from the lamp. Of all the light emitted by the lamp the fraction that arrives directly is approximately ##\frac{\pi y^2}{4\pi x^2}=(\frac y{2x})^2##.
The light arriving on the same area after reflection is the light that falls inside a circle radius y on the mirror. If this light was emitted in a cone of half angle ##\theta## then ##\sin(\theta)=\frac{4fy}{y^2+4f^2}##, where f is the focal length. As
@Dale noted, that is a solid angle ##2\pi(1-\cos(\theta))=4\pi\frac{y^2}{y^2+4f^2}##, so of all the light the lamp emits the fraction arriving on the target by reflection is ##\frac{y^2}{y^2+4f^2}##.
To get the total light on the target, add these fractions together.
Edit: more thoughts...
The analysis above only gives illumination in an area up to the aperture of the mirror. Beyond that, no reflected light hits the target. It looks like you want to brighten an area significantly larger than the aperture of the mirror, so it might be better to position the lamp somewhere between the focal point and the apex. Or use a different shape of mirror.