  Parallel plate capacitors (PPC) significantly reduce the size of
superconducting microwave resonators, reducing the pixel pitch for arrays of
single photon energy-resolving kinetic inductance detectors (KIDs). The
frequency noise of KIDs is typically limited by tunneling Two-Level Systems
(TLS), which originate from lattice defects in the dielectric materials
required for PPCs. How the frequency noise level depends on the PPC's
dimensions has not been experimentally addressed. We measure the frequency
noise of 56 resonators with a-SiC:H PPCs, which cover a factor 44 in PPC area
and a factor 4 in dielectric thickness. To support the noise analysis, we
measure the TLS-induced, power-dependent, intrinsic loss and
temperature-dependent resonance frequency shift of the resonators. From the TLS
models, we expect a geometry-independent microwave loss and resonance frequency
shift, set by the TLS properties of the dielectric. However, we observe a
thickness-dependent microwave loss and resonance frequency shift, explained by
surface layers that limit the performance of PPC-based resonators. For a
uniform dielectric, the frequency noise level should scale directly inversely
with the PPC area and thickness. We observe that an increase in PPC size
reduces the frequency noise, but the exact scaling is, in some cases, weaker
than expected. Finally, we derive an engineering guideline for the design of
KIDs based on PPC-based resonators.
