Yes, but interestingly, even on offense, the Union’s casualty rates were lower than Lee’s-—7 Days’ Battles, for example. And I’m not even throwing in the overwhelmingly lopsided surrender of Vicksburg to Grant, or Ft. Donelson. In those, the Confederacy lost 100% of its men deployed, and, no, Lee wasn’t in charge of either.
The battle was fought in central Virginia over the same ground as the Battle of Gaines' Mill during the Seven Days Battles of 1862. In fact, some accounts refer to the 1862 battle as the First Battle of Cold Harbor, and the 1864 battle as the Second Battle of Cold Harbor. Soldiers were disturbed to discover skeletal remains from the first battle while entrenching. Despite its name, Cold Harbor was not a port city. It described two rural crossroads named for a hotel located in the area (Cold Harbor Tavern, owned by the Isaac Burnett family), which provided shelter (harbor) but not hot meals. Old Cold Harbor stood two miles east of Gaines' Mill, New Cold Harbor a mile southeast. Both were approximately 10 miles (16 km) northeast of the Confederate capital of Richmond
Check out this link: Civil War battle statistics
Note that Lee's casualties were sometimes, but not always, higher than his Union opponent's.
I think a bit of analysis would ask: who was on offense, who was on defense, and who won the battle. The results could well explain the differences in casualties.