# Error associated with parameters in Minimize (or NMinimize) in discrete fitting

my problem goes as follows:

I need to a discrete fit of a discrete function. Say I have the following made up data points, which are generated by the parabola f[x_]:=x^2 with some "experimental error" added:

data := {{-3, 8.9}, {-2, 4.1}, {-1, 1}, {0, 0}, {1, 1.2}, {2, 3.9}, {3,9.2}};
valuex := {-3, -2, -1, 0, 1, 2, 3}
valuey := {8.9, 4.1, 1, 0, 1.2, 3.9, 9.2}


These points should be fitted by a curve of the general formula f[x_]:= a*x² + b*x + c, and I built a NMinimize algorithm as:

Minimize[Sqrt[Plus @@ ((a*valuex^2 + b*valuex + c - valuey )^2)], {a, b, c}]


This gives off a pretty accurate value for a, b and c

{0.282, {a -> 0.99881, b -> 0.025, c -> 0.0476191}}


these are basically the same values I got when using NonlinearModelFit

nmf = NonlinearModelFit[data, a x^2 + b x + c, {a, b, c}, x]
FittedModel[0.047619+0.025x+0.99881x^2


Now what nmf gives me, that I need to calculate in the NMinimize case is the nmf["ParameterErrors"], {0.0153843, 0.0266465, 0.0814063}.

How can I do that manually? I tried looking it up how NonlinearModelFit does it, but my lack of mathematical knowledge is making things pretty hard.

Any help regarding that matter is welcomed!

p.s.: I don't know if I made it clear enough, but that is just an example function. The one I need to fit is much more complicated and cannot be analytically evaluated, making NMinimize essential.