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Using a 3D general circulation model (GCM), we investigate the sensitivity of the climate of tidally-locked Earth-like exoplanets, Trappist-1e and Proxima Centauri b, to the choice of a convection parameterization. Compared to a mass-flux convection parameterization, a simplified convection adjustment parameterization leads to a $>$60% decrease of the cloud albedo, increasing the mean day-side temperature by $\approx$10 K. The representation of convection also affects the atmospheric conditions of the night side, via a change in planetary-scale wave patterns. As a result, using the convection adjustment scheme makes the night-side cold traps warmer by 17-36 K for the planets in our simulations. The day-night thermal contrast is sensitive to the representation of convection in 3D GCM simulations, so caution should be taken when interpreting emission phase curves. The choice of convection treatment, however, does not alter the simulated climate enough to result in a departure from habitable conditions, at least for the atmospheric composition and planetary parameters used in our study. The near-surface conditions both in the Trappist-1e and Proxima b cases remain temperate, allowing for an active water cycle. We further advance our analysis using high-resolution model experiments, in which atmospheric convection is simulated explicitly. Our results suggest that in a hypothetical global convection-permitting simulation the surface temperature contrast would be higher than in the coarse-resolution simulations with parameterized convection. In other words, models with parameterized convection may overestimate the inter-hemispheric heat redistribution efficiency.