I was wondering, does the level of the parents and their stats affect the level or stats of the offspring?

Yep, both these things affect the offspring.

If an offspring has high-level parents, it may hatch out at higher than Level 0. Some of my pets have been Level 5 just as soon as they hatched! This gives you a head start on leveling up that pet when it is old enough. However, the offspring won't go higher than the same max level as all other pets, which is currently Level 60.

On the other hand, high-stat parents have high-stat offspring, and this benefit lasts. Suppose two pets hatch out, identical to each other except that one had parents with base level stats and the other had parents with higher level stats. The second one will have higher stats than the first one will when the two of them hatch out. Furthermore, when these two offspring reach Level 60, the second pet will still have higher stats than the first one, and the first one will never, ever catch up.

The maximum possible stats at hatching are Health 2,000 and Strength, Intelligence, Agility, and Dexterity at 100 each. This is called a "Max Stat Pet." These are valuable pets; I own one. Here he is if you'd like a look--


"Piccolo"

Okay great! Thank you for your help! :)

BTW, "Tuatha De Danaan" is a neat name for a pet.

If you want some high-stat pets, now is a good time to buy them. Lots of people are mass-breeding pets just so they can Nurture the eggs and Hatchlings for the Spring Festival Nurturing contests. Many of these pets are not very pretty but have high stats. The players often don't list the pets for sale, but a lot of players will sell them cheap or even give them away if you politely ask to acquire the pet. I just got these two babies--


Zachary and Kali

Zachary's stats are high for a Level 0: Health 254, Strength 27, Intelligence 39, Dexterity 23, Agility 34.
Kali's are even higher: Health 389, Strength 56, Intelligence 43, Dexterity 39, Agility 36.

I'm posting them here so you can get an idea of what you can acquire at this time. The thing is, you have to find the Hatchlings and then ask to buy them. These two were just two of about a hundred Hatchlings with ordinary to very high stats in one Hatchery.

Good luck!