If he's not waking up in the night, he should be getting up early in the morning every morning.
Really, it would probably be fairest if you alternate- so one night you do the wake-ups and he does the morning, then switch for the next day. But tbh, I ended up doing the night wakings because my DH was slower to hear DS than me, and I'm such a light sleeper anyway that I'd be awake no matter how speedy he was. So, I got the lie in every morning, and he'd handle DS right up until he went to work.
That was actually fine with me because I found I slept better when I at least had some kind of pattern, in that I knew I could sleep uninterrupted from the last night wake to when DH went to work. Body clock just adjusted in the end- DH got used to going to sleep earlier and waking up earlier, and I got used to going to sleep later and waking up later. I think you sleep better if you sleep and wake at roughly the same time every day, although I know the night-on/night-off pattern works fine for many.
Basically, I think both parents should be entitled to the same amount of sleep. However you arrange that to happen probably doesn't matter too much as long as you both get enough sleep to keep ticking over. If he's getting every single night with no wake-ups AND a lie-in every other day, that's not bloody on.