Charging your laptop battery generates heat. So when you plug in your AC power, and your laptop battery isn't fully charged (it's previously been running off battery), it will start charging the battery, thus generating heat.
This is why you usually get the fan spinning up when switching to AC power after being on battery, and inevitably the system components will heat up due to the heat from the battery.
The hard drive, being reasonably close to the battery, will probably heat up a bit too. This therefore is a plausible explanation of what you are observing.
There are other things that would cause your laptop to get warmer when it's on AC power, such as various components coming out of a "low power" mode, but the battery charging, IMHO, is a much more significant factor.
You say that your hard drive is "overheating" but only mention that it reaches 60 degrees. In reality, manufacturers usually list 60 degrees as the top of the "safe operating temperature" for 2.5 inch drives. Your laptop manufacturer will have designed your laptop's cooling system so that the most extreme situation (charging battery while stressing CPU, etc) should still cause all components to remain within their safe operating temperature. However, they make a number of assumptions. They assume that your room temperature is a certain value, that you're using the laptop on a solid, flat surface, its ventilation holes aren't blocked, etc. If it's 28 degrees inside and you have the laptop on your lap, then components will exceed their safe operating temperature. Similarly, if the fan fails or is blocked, then it will also.
It's also not unusual for laptops simply to run too hot because they have poor thermal design from the start.
The hard drive manufacturer will have given some "breathing room" in specifying 60 degrees as the top safe operating temperature, which will partially make up for non-ideal cooling situations. That is, if you go up to 65 it shouldn't be "too bad". You probably don't need to worry about imminent death of your laptop. But any heat will reduce the life of your components and the cooler the better. You should look into whether the fan is operating correctly, whether you're using the laptop in a hotter environment than a typical room temperature, and whether you have the laptop on a soft or warm surface or one which will press against the air vents. If using the laptop on your lap, get a solid tray/stand for it to sit on. Some companies sell these with built-in cooling fans which I think is overkill.