Can the parallel capacitance of a quartz crystal be directly measured?

Short answer: measure $C_p$ at a frequency far from any one of the crystal's resonant frequencies.
Be aware that not only is a crystal resonant on harmonics of its primary (printed) frequency, but spurious resonances also appear.

It is probably safest to measure the parallel-plate capacitance below resonant frequency. So yes, LCR bridges often use a signal source at low frequency. Since $C_p$ may be in the single-digit picofarad range, care should be taken to reduce stray capacitance of the measurement fixture.
You might measure $C_p$ with the crystal mounted in the LCR meter, then carefully remove the crystal without disturbing the environment, and measure stray capacitance. The true(er) $C_p$ is the difference between the two measurements.

For quartz AT-cut crystals, $C_p$ is very roughly 250 times larger than $C_s$.
You're unlikely to measure $C_p$ so accurately in a LCR meter or bridge that this factor need be considered.

---

Ceramic resonators are not as piezo-active as quartz, so the disturbing effect of $C_s$ requires factoring-in its effect on a $C_p$ measurement.
For example, an ultrasonic transducer was measured where $C_p$ was only 7.5 times larger than $C_s$.
Quartz has amazing qualities: very piezo-active, temperature-stable. Most other materials that are piezo-active pale in comparison.

The impedance of $C_p$ is

$$ Z_p(omega) = {1 over i omega C_p } $$

and the impedance of the other branch is

$$ Z_s(omega) = R + i omega L + {1 over i omega C_s} $$

and the impedance of the entire crystal is the parallel combination of these:

$$ Z_x(omega) = left( {1over Z_p(omega)} + {1over Z_s(omega)} right)^{-1} $$

Typical values for an approximately 14 MHz crystal are:

$$begin{align} R &= 7.37 :Omega\ C_s &= 18.8:mathrm{fF}\ C_p &= 4.15:mathrm{pF}\ L &= 6.57:mathrm{mH} end{align}$$

At 1.4 MHz:

$$ begin{align} Z_p(2pi 1400000) &= 0-27393i\ Z_s(2pi 1400000) &= 7.37 - 5989128i\ Z_x(2pi 1400000) &= 0.000152779-27269i end{align}$$

An LCR meter is probably just measuring the magnitude of the voltage when applying a known AC current, so it will be off by a factor of:

$$ {|Z_x(2pi 1400000)| over |Z_p(2pi 1400000)|} = 0.995447 $$

So for this particular crystal, measuring the impedance with an LCR meter (assuming no other inaccuracy in the device) at approximately 1/10th the serial resonance frequency of the crystal yields an error of less than 1%.

So, given an LCR meter that can be accurate with a reactance on the order of 30kΩ, this is not a bad way to go.