well that would be me with the answer that turned out to be less helpful, so let me flesh it out a bit...
 
Different types of equalizers behave differently, and there are a LOT of variations. Filters in series vs filters in parallel, compensation for summing vs no compensation, IIR vs FIR (ok, only in the digital domain), and of course parametric vs multi-fixed-band, aka graphical EQ.

Even within the world of graphical EQs there are different filter types and widths and so on.

So my answer was quite serious, albeit apparently too brief. Sometimes a parametric filter is what I need, and sometimes a graphical EQ is what I need. And really, that's how I decide - but that's based on starting out in the mid 1970s when most console filters (if they were even included) were fixed parameter filters, and outboard usually meant a graphical EQ or two.

Two of my all time favorite equalizers are both graphical - which even as I type this fascinates me a bit. The best sounding filter set I've used is the original API 560 - 10 bands of musical goodness! And I think that probably influenced my ears a bit, because my other favorite was made by  Traynor back in the 1980s, it could be a 31 band 1/3 octave filter set, or two 10 band 1 octave filters sets. I used it mostly in that second configuration.

I still love parametric equalizers, and my favorite in that bunch is the Valley Audio Maxi-Q. But the API, Ashly, Neve, Tangent, and Trident parametric equalizers are also on the list, actually, it's a long list, probably easier to list the ones I don't like, but I'll skip that.
 
I think if I had to have a rule of thumb it would go something like I use graphical equalizer more often when I am trying to shape a track, or even a mix. I use parametric equalizers when I am trying to solve problems. But sometimes a parametric will let me shape the track better than the graphic. I can't think of occasions when the graphic solved a problem better than a parametric - but I'm sure there are cases.
 
As far as using any filter set based on a graphical user interface I would have to say that seldom comes up for me. I find that using my eyes for make audio decisions just doesn't work for me. I'm in awe of people that can edit or set parameters for a filter or compressor by eye. I've been in the room when it has happened, so I don't discount it, I just can't do it. I get better results using my ears, and in fact when I was schlepping coffee and take-out my 'teachers' would often cover the controls to make me use my ears.

It was probably helpful at one time - but frankly the numbers around the knobs are a great starting point!
 
So that's the long(er) answer - I use the tool that gets me the result I'm listening for. And as often as not these days I have a pretty good idea of what  I am trying to do. I've also developed a reasonably good idea of how to use my microphone locker - selection and placement - to make the need for filters and compressors less likely.

One last thought - I have one parametric EQ - and Ashly but I don't know the model, it was part of a custom rig - anyway, when I set the bandwidth really tight the filter will ring on transients. It's a cool effect!