You could always fill in the gaps with amphibian DNA.

> Is it possible to use deep learning to fill out the missing parts?

The half life is 521 years - there is nothing left to fill in. If you read the linked article you'll see even under perfect conditions all DNA is destroyed after about 6 million years. This fossil is 110 million years old.

If you use deep learning, the missing parts will be filled with something based on whatever you trained the model on. In other words, the training data won't be dinosaur DNA as we don't have that. At best, during training you can classify some DNA as "more dinosaur" and other DNA as "less dinosaur" and you hope the result will be some extrapolation towards "most dinosaur". Despite the practical limitations, it's certainly an interesting thought.