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Data from the space missions {\it Gaia}, {\it Kepler}, {\it CoRoT} and {\itTESS}, make it possible to compare parallax and asteroseismic distances. Fromthe ratio of two densities $\rho_{\rm sca}/\rho_{\pi}$, we obtain an empiricalrelation $f_{\Delta \nu}$ between the asteroseismic large frequency separationand mean density, which is important for more accurate stellar mass and radius.This expression for main-sequence (MS) and subgiant stars with $K$-bandmagnitude is very close to the one obtained from interior MS models byY{\i}ld{\i}z, \c{C}elik \& Kayhan. We also discuss the effects of effectivetemperature and parallax offset as the source of the difference betweenasteroseismic and non-asteroseismic stellar parameters. We have obtained ourbest results for about 3500 red giants (RGs) by using 2MASS data and modelvalues for $f_{\Delta \nu}$ from Sharma et al. Another unknown scalingparameter $f_{\nu_{\rm max}}$ comes from the relationship between the frequencyof maximum amplitude and gravity. Using different combinations of $f_{\nu_{\rmmax}}$ and the parallax offset, we find that the parallax offset is generally afunction of distance. The situation where this slope disappears is accepted asthe most reasonable solution. By a very careful comparison of asteroseismic andnon-asteroseismic parameters, we obtain very precise values for the parallaxoffset and $f_{\nu_{\rm max}}$ for RGs of $-0.0463\pm0.0007$ mas and$1.003\pm0.001$, respectively. Our results for mass and radius are in perfectagreement with those of APOKASC-2: the mass and radius of $\sim$3500 RGs are inthe range of about 0.8-1.8 M$_{\odot}$ (96 per cent) and 3.8-38 R$_{\odot}$,respectively.

Fundamental properties of stars from Kepler and Gaia data: parallax offset and revised scaling relations