Can a parent purchase a handgun for their child who is under 21?

Under Federal law, people under 21 cannot buy a handgun from an FFL, but they can buy one from a private individual. Also, people under 21 can be gifted a handgun. If your parents buy you a gun, and you pay them for it, that is a straw purchase regardless of whether you could have legally bought the gun yourself. If your parents already have the gun, and you buy it from them, that's ok. To buy a gun from someone in another State, they have to ship it to an FFL in your State.
Check on the laws of Maryland regarding handgun possession.

I wouldn't rely on this forum for an answer as it's solidly a state law question best answered by the State AGs office. Also, Federal law prohibits transfer to someone under 18. Some exceptions exist under 18 U.S.C. § 922(x)(2), (5), so you want to review that as well. There's stuff about juveniles having weapons for employment (e.g. ranches), transporting, etc. One issue that has caused the regulations and case law to get tinkered with is the concept of "constructive possession". Go over that as well. I'm not one to give advice on what's OK and not, but at least I can give you a few things that you need to look into. Straw purchases become a big topic when something illegal happens involving a juvenile and weapons. Good to ask the AG office. My best real advice is to ask at the source and stay well clear of the legal/illegal line.

I am not sure from your post what state (Maryland or Ohio), you plan on making this purchase. Ask your question at a reputable gun shop. Thy will not sell you a handgun that would jeopardizes their Class III license. You can also call the State Firearms licensing division and ask them the same question. Good luck Sir.