If no frame of reference has precedence over any other, why do we say that if a spacecraft leaves earth and attains a significant percentage of c, time would pass more slowly for the crew of the ship than for those still on earth?

The crux here is not in the relative velocity but in the acceleration of one of the objects. There is a "preferred" frame if one of the objects are accelerating. To a third observer, it could appear that two objects are accelerating away from each other, but only one object feels the force of the acceleration, the one with the rocket engines blazing.

Relative to an unaccelerated frame, an accelerated frame experiences less time passage. This is true if the acceleration is from a propulsive engine and/or if the acceleration is from the local curvature of space (gravity).

This is the best link I could find on short notice:
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html#c4
For a constantly accelerating rocket, you could replace g in the equation with a, the acceleration of the rocket. It's more complicated otherwise and over my experiential horizon.

BTW, If anyone want's to dispute the fact that the concept of relativistic mass is a seriously flawed idea, I'm ready to rumble.