Yoshi got dragged into Peachgate entirely by accident. If the poison princess hadn't chosen his native habitat for a sordid rendezvous with Bowser in 1990 he'd still be living a perfectly innocent and happy life. But we all know what happened. Unaware of the backstory to the whole depressing affair, Yoshi agreed to help Mario on his never-ending quest for emotional buggeration and forged a strong friendship with him in the process, despite the obviously crippling back pain that comes from having a fat plumber on your back for hours at a time. You wouldn't believe how sore that gets. (We've been told)
Later on however, presuambly through conversations with Luigi during the first Super Mario Kart, Yoshi found out what was going on. He continued to hang out with Mario for a while, still feeling the bond of their shared journey and sympathising with the hard start in life the plumber had had on his very own island as a baby. Having a man's arse imprinted upon your person kind of makes you feel pretty close to him too. (We've been told)
But eventually Mario's lack of self-esteem got too much and Yoshi followed Luigi's example. Thus he went straight to the roof when Peach engineered a disappearance in Super Mario 64, and resolutely stayed there until Mario had finished humiliating himself. And he didn't even bother turning up for Super Mario Galaxy. What does that tell you?