I didn't realize this for a few days but pets don't age if the day ticks over while they're exploring or if you haven't brought them back from Relking Coin hunting and the day ticks over. I think they should still age if this is the case. I only tested it on a whim but I know that there have been a few times my pets have been out at midnight Res time and I'm pretty sure I've lost SEVERAL days of age with them.

If that's not possible for some reason, there should be a warning somewhere about it because I certainly had no idea. I'm only glad I don't use my oldest pet to do the exploring.