All drives (HDD and SSD) are always active to some extent. I have one machine left with a HDD (everything else is SSD). The HDD machine is near silent.
Most machines with Hard Drives (Customer machines) are near silent.
I have one old customer machine with a slow 5400-rpm HDD that makes a muffled sound when writing big amounts.
So 90% or more of my machines and customer machines are near silent in normal operation.
Keep that HDD of your unplugged unless you need it. Some drives (fewer than more) are noisier than others. The high level of noise can be a forerunner of failure also. Consider replacing the drive with an SSD drive. Good quality (Samsung) SSDs have long life spans.
With Indexing, Search, update checking and so on, it is near impossible to stop a disk from reading / writing, so just unplug it.