Measuring parasitic capacitance of coil using an LCR meter
A regular LCR bridge measures the dominant net impedance. That measured impedance will be inductive if the frequency the bridge uses is below the self-resonant frequency (SRF) of the coil. But, the closer the bridge's test frequency is to SRF, the bigger the error because the parasitic capacitance is starting to become dominant.

If the bridge uses a frequency above the self resonant frequency of the coil then it will indicate a value of capacitance that is somewhat representative of the parasitic capacitance. The higher the frequency is above SRF the better the reading is at representing the true parasitic capacitance.

If your bridge can use multiple frequency settings for measuring impedances then make use of this and, using the standard resonant frequency formula you can estimate the parasitic capacitance.