Why use parametric based estimation methods - confusion regarding terms
Hi: I'll try to answer as briefly as possible and only with respect to statistics. not dsp. 

In statistics, if you have a nice pdf such as the normal distribution, then maximizing the likelihood is equivalent to minimizing the sum of squares of the residuals ( often called errors ).

In other cases, where you either have a complicated distribution ( maybe the model is non-linear and results in a more complex likelihood), then maximizing the likelihood is not equivalent to minimizing the residual sums of squares. So, you may still choose to maximize the likelihood but because the likelihood is complex, one often needs to turn to a numerical optimization algorithms such as gradient descent, BFGS.

So, I think your confusion is due to the fact that it ( maybe unfortunately ) happens to be the case that maximizing the likelihood of the normal is equivalent to minimizing the residual sums of squares. 

Then there are other cases maximizing the likelihood is too difficult or the likelihood is not even derivable in closed form.  In this case, people define other objective functions ( heuristics ) that still measure how "well" the model fits the data. Doing that, allows one to use the genetic, particle swarm type optimization schemes that you refer to. These schemes are often called non-parameteric because these objective functions are not tied to a parametric likelihood ( such as the normal,t, gamma etc ). In fact, the likelihood is often not even known.

Basically, what type of optimization one uses depends on whether you have a pdf and how complicated it is. 

A reallly rough categorization is as follows:

1) normal case: : straightforward: maximize it or minimize residuals squared.

2) non-normal case: likelihood is complex, maybe non-linear but known  : bfgs or gradient descent or some other numerical method that can handle the 
closed form version of the likelihood.

3) likelihood too complex or unknown or doesn't exist: define one's own objective function that measures the quality of the model and use algorithms such as particle swarm etc.

Note that in 2) above, you have to worry about whether the optimum you hit is local, global etc and this can be extremely tricky.
This may be the case in 3) also. I'm not as familar with genetic algs as the numerical methods.

I hope this helps some. I'm not sure of a good reference for what you asked about. Kenneth Lange has a book "numerical methods in statistics' that may be relevant but I don't have it. Good luck.