At the equinox the earth rotation axis is in a plane that is tangential to the planet orbit, and orthogonal to the orbital plane. Hence it cannot be tilted towards the sun or away from it.
But it does not mean that it is not tilted, but simply that (assuming the axis has an orientation) it is tilted forward, or backward, on its path around the sun. And since it is tilted, the celestial equator and ecliptic are not parallel.
By the way, if it is oriented towards the sun on one solstice and away on the other, it has to go through intermediate positions. Actually, the axis keep approximately the seme orientation (with respect to stars) all year long, up to precession.