Aims. We present a comparison of Gaia Data Release 1 (DR1) parallaxes with photometric parallaxes for a sample of 212 Galactic Cepheids at a median distance of 2 kpc, and explore their implications on the distance scale and the local value of the Hubble constant H0. Methods. The Cepheid distances are estimated from a recent calibration of the near-infrared period-luminosity (P–L) relation. The comparison is carried out in parallax space, where the DR1 parallax errors, with a median value of half the median parallax, are expected to be well-behaved. Results. With the exception of one outlier, the DR1 parallaxes are in very good global agreement with the predictions from a well-established P–L relation, with a possible indication that the published errors may be conservatively overestimated by about 20%. This confirms that the quality of DR1 parallaxes for the Cepheids in our sample is well within their stated errors. We find that the parallaxes of 9 Cepheids brighter than G = 6 may be systematically underestimated. If interpreted as an independent calibration of the Cepheid luminosities and assumed to be otherwise free of systematic uncertainties, DR1 parallaxes are in very good agreement (within 0.3%) with the current estimate of the local Hubble constant, and in conflict at the level of 2.5σ (3.5σ if the errors are scaled) with the value inferred from Planck cosmic microwave background data used in conjunction with ΛCDM. We also test for a zeropoint error in Gaia parallaxes and find none to a precision of ~20 μas. We caution however that with this early release, the complete systematic properties of the measurements may not be fully understood at the statistical level of the Cepheid sample mean, a level an order of magnitude below the individual uncertainties. The early results from DR1 demonstrate again the enormous impact that the full mission will likely have on fundamental questions in astrophysics and cosmology.