I find that the advantage of a parametric EQ is that it can address both fine and specific adjustments as well as more sweeping and general adjustments better than a fixed bandwidth and frequency graphic EQ. A parametric equalizer can not only address narrower bandwidths and be centered at any frequency but they can also address a wide bandwidth, low Q change without the band interaction, etc. that might be involved on a graphic EQ. The challenge is that parametric equalizers can be more difficult to apply and unless accompanied by related software that shows the resulting equalization graphically, are not as visual in nature.I really depends on the situation. I feel in general, the use of a parametric EQ gives you greater flexibility in situations like yours where you are talking about fine and specific adjustments, as opposed to sweeping curves.
A point that many people don't realize is that in DSP implementation a graphic EQ is often simply a 'macro' of a large number of parametric filters with prefedined center frequencies and bandwidths. So a 31 band graphic EQ in a DSP or digital console is effectively 31 parametric filters with defined center frequencies and bandwidths and for which you can adjust only the amplitude. This was a significant factor in early DSP devices as a one-third octave graphic EQ required four to six times the processing of a 5 to 7 band parametric equalizer and that was typically a significant portion of the processing power available. You still see this being a factor in some entry level DSP devices and digital consoles where the number of graphic equalizers available may be limited as they require significant processing resources.