  Gravitational-wave observations provide a unique opportunity to test general
relativity (GR) in the strong-field and highly dynamical regime of the theory.
Parametrized tests of GR are one well-known approach for quantifying violations
of GR. This approach constrains deviations in the coefficients of the
post-Newtonian phasing formula, which describes the gravitational-wave phase
evolution of a compact binary as it inspirals. Current bounds from this test
using LIGO/Virgo observations assume that binaries are circularized by the time
they enter the detector frequency band. Here, we investigate the impact of
residual binary eccentricity on the parametrized tests. We study the systematic
biases in the parameter bounds when a phasing based on the circular orbit
assumption is employed for a system that has some small residual eccentricity.
We find that a systematic bias (for example, on the leading Newtonian
deformation parameter) becomes comparable to the statistical errors for even
moderate eccentricities of $\sim 0.04$ at $10$ Hz in LIGO/Virgo band for binary
black holes, and $\sim 0.008$ for binary neutron stars. This happens at even
lower values of orbital eccentricity in the frequency band of third-generation
(3G) detectors like Cosmic Explorer ($\sim 0.005$ at $10$ Hz for binary black
holes and $\sim 0.002$ for binary neutron stars). These results demonstrate
that incorporating physical effects like eccentricity in waveform models is
important for accurately extracting science results from future detectors.
