The sentence you wrote is almost correct but needs a small adjustment to make it clearer and grammatically correct.
Here is your original sentence:
The penguin dad gently holds the egg on his feet tucked away in his feathers. While the dad keeps the egg warm, the mother is off swimming and catching fish for her family.
Correction:
"The penguin dad gently holds the egg on his feet, tucked away in his feathers. While the dad keeps the egg warm, the mother is off swimming and catching fish for her family."
Explanation:
We add a comma after "feet" to separate the phrase "tucked away in his feathers," which describes how the egg is held. This makes the sentence clearer and easier to understand.
So, the corrected sentence tells us that the dad penguin is carefully keeping the egg safe and warm on his feet, while the mom penguin goes to find food for the family.