Measuring small capacitor parallel to big inductance (Network Analyser)
Set up your network analyzer to plot the response as magnitude of impedance (\$\left|Z_L\right|\$), with log scale on the y axis. 

Assuming 1. your network analyzer can reach a high enough frequency, and 2. the connection from the reference plane to the termination you're trying to measure is short, you should see \$|Z_L|\$ increasing from low frequencies up to some peak and then decreasing after the peak.

The equation for the part of the curve where it's increasing is approximately \$Z_L(f)=j2\pi{}fL\$. The equation for the part of the curve where it's decreasing is approximately \$Z_L(f)=\frac{1}{j2\pi{}fC}\$. By picking out a couple of points on the appropriate parts of the curve, you should easily be able to estimate both \$L\$ and \$C\$.

If you already know \$L\$ you can get \$C\$ really easily knowing the resonance frequency (the frequency where the graph reaches its peak) is \$f_R=\frac{1}{2\pi\sqrt{LC}}\$.