The side that would make war would be the South or the Confederacy. The side that would accept war would be the North or the Union.
The full quotation is, "Both parties deprecated war, but one of them would make war rather than let the nation survive and the other would accept war rather than let it perish, and the war came."
Lincoln attempted to lay blame at the feet of both sides of the issue. Both parties or groups disliked war, but were so drawn to it based on their perspectives of why their side was justified and why the other side was wrong. He was attempting to distribute blame, but his speech was criticized and misunderstood at the time.