To popularize the asteroid theory of dinosaur extinction,

I used the below reasoning:

Upon reading the original statement carefully, I can make out that the sentence is a suggested action that could be taken if a certain reaction is to be achieved.
Now, let's see the splits in the options:
1) have to/would have to
2) link/linking

My first elimination is on the basis of the first split. The sentence tries to bring out a suggested action that can be taken for the result. Using 'Have to' in this case is incorrect, because it conveys a necessity as opposed to a hypothetical situation which the sentence is deliberating. Hence, 'would have to' is correct.

Options A and B are eliminated

Option E is eliminated as its not parallel. "Paleontologists would have to <do something>"..than.."now".

Between options C and D:
The sentence conveys the idea that the evidence is question is fact linking "the disappearance of fossil records and the layer of sediment in the Earth's crust caused by an asteroid impact". It is not a separate task that needs to be done. Hence, 'linking' is the right usage

Option D is correct