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 ∼0.04 at 10 Hz in LIGO/Virgo band for binary black holes, and ∼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 (∼0.005 at 10 Hz for binary black holes and ∼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.