Your father climbed to some rough rocks near the coast to find out that under the rocks, our friend Lake lies severely wounded.
Is this usage of "to find something by chance (as a result of climbing)" correct?
closed as off-topic by FumbleFingers, Josh61, Mari-Lou A, Zairja, Rory Alsop Aug 26 at 9:05
The way it is written could be improved. As it stands it does suggest that 'your father' happened upon the discovery by chance. But that is largely due to the circumstances of this particular account.
Consider for a moment He looked up to see a dodo flying backwards.
It could mean either that he had just been told there was a dodo flying backwards, so he looked up to see it. But it could also mean that he looked up (for other reasons) and lo and behold there was a dodo flying backwards.
To be absolutely clear that something was discovered by chance you need to say something like:
He looked up, and what should he see but a dodo flying backwards. You can replace 'and what should he see' with many other phrases, such as 'to his great surprise', 'to his utter astonishment', etc.
You can also say He looked up, only to see a dodo flying backwards. This last does convey the idea that it was a discovery on his part. It would be a good way of dealing with your example:
Your father climbed to some rough rocks near the coast, only to find that under those rocks our friend Lake lay severely wounded