What are some of the disadvantages of parametric PSD estimation methods?
Can we trust a spectrogram obtained this way even though the signal can't be modelized or doesn't fit the correct framework of the used method ?


Short answer: anything based on a model will work only when applied to the model. Not quite sure which other answer you were expecting here!

Of course, some "model mismatches" are more benign than others: For example, some classes of estimators are very robust against noise not fitting the original model, but break down when the assumed number of signal components isn't what was specified (example: ESPRIT). Others fail the other way around (Bartlett's method will not like you having negative SNR).