There was a time when the position and orbital parameters of objects within the solar system (and, indeed, without) were given in reference to the ecliptic plane and vernal equinox.
However the vernal equinox moves pretty quickly, and even the ecliptic plane moves, so astronomers and other off-earth enthusiasts now use the ICRF. This was first established in 1998 and significantly updated and improved in 2009 (as ICRF2).
The ICRF2 does not make reference to the ecliptic, the celestial equator, the equinoxes, but is instead based on very long base line interferometry of hundreds of extremely distant extragalactic sources (mainly, quasars), whose combined movement from year to year, or indeed from millennium to millennium, is not significant in comparison to the precision required for this purpose.
The coordinate system based on the ICRF2 is called the ICRS (International Celestial Reference System). The axes of the ICRS do roughly match the axes of the ecliptic-equinox based systems (as at year 2000), but are basically fixed (inertial). The gap between the ICRS and the ecliptic and equinox will increase over time.
Orbital elements for solar system objects that you encounter generally are likely to refer to the ICRS.
The ICRS is centred on the barycentre (centre of mass) of the solar system rather than the centre of the sun. The position of the barycentre with respect to the sun depends largely on the relative positions of the gas giants. The xy plane of the ICRS is inclined by about 1.5 degrees with respect to the invariable plane: this is the plane perpendicular to the total angular momentum of the solar system (which again is mainly influenced by the sun and the gas giants).