As far as I'm aware, you can only lend a pet by entering one of the sigils in the Hatchery. Once you do so, two options will pop up: hatch a pet (costs gold & uses up timer, but you get a egg), or lend a pet (doesn't cost gold or use up hatching timer, but you don't get an egg). Then you just select whichever option you want, and you're good to go!