Do you consider the term jarhead and other names for the different branches of service offensive?

No as already mentioned all in the military family. Marines and sailors were always using slang names in reference to each other. Much slang of other services started in Parris Island with more cussing and vulgariety than previously thought possible. Aboard ship the sea, air, land types were questioned, how long can you balance a ball on your chin anyway ? Then there were the seal sounds and clapping hands. Deck ape, squid, paint chipper, anchor clanker, pixie cup referred to the head gear. Navy gave it right back to the grunts on their vessel with a colorful cussing. Lining up for ship chow the ice cream machine would break.
Recent events made BAM a bad reference. Fifty plus years ago it was used not in a derogatory manner. Just a nickname term of endearment for the WM.
The U.S. Air Force always treated a Marine well. They did not care for Marines off the flight line with weapons. In a uniform with no chevrons got me into a U.S. Air Force Officers Club Hickam Transport across the the base provided by a U.S. Air Force Police butter bar lieutenant. Air dale, wing wipper and odd references to being civilians were still used. No better chow or base services than U.S. Air Force.
Doggie, dog face, ice cream soldier, ricky ranger and SF were not spared. U.S. Army chow better than Marine.
Coast Guard then under the Deparment of Transportation. Duck ass navy, knee deep navy and those guys lived on a yatch. All in the family.

I try not to use the term Jarhead to describe a Marine. I believe crayon eater to be more accurate.

No, why?

Interservice rivalries are, in my opinion, an important part of the camaraderie of the branches. 99% of Marines won't care if you call the jarhead as long as you're not clearly trying to be a jerk. Just don't call the 'Sarge' Marines really don't like that.

call me anything but late for dinner... I was Army, so admittedly I'm a little fond of the term "knucklehead"

What does it matter if it comes off as offensive? That's why it's called rivalry. If you get a little stressed because some jarhead calls you a name...how are you gonna react to combat?

I like to remind them that submarines are a myth. After all, there is nothing lower than a Marine! ba dum bum, tish! But I'd never accept that from a civilian!

Jarhead is a derogatory term. It's the same as a white person calling a black person a n****r. However, if you are what you call someone else, then it's meaning isn't quite the same.

While I served on the MarDet on the U.S.S. Midway... I had a squid friend of mine once tell me the word Marine is an acronym.... Standing for Muscles Are Required Intelligence Not Expected... I was laughing so hard, I could not beat his ass for it. Now, I have a good friend who is (get this) Retired Coast Guard, he and I bust each others balls constantly. And I agree, we are all family. We have the right to bust each others chops...

It never bothered me or my dad.