How old is the laptop? If it’s really old or it has been used every single day for hours, then you might need to do laptop maintenance, which includes blowing out dust from the fans and replacing thermal paste.
I think the easiest ways of all to check if the HDD is running is to start some write-intensive task and then put your ear onto the keyboard. If you hear the common Hard Disk noise such a device does, when it is running, then it’s turned on. However, sometimes it’s hard to hear, especially with quieter hard drives.
It also can happen that your HDD is mainly off, but may be started during the boot process, for example, if GRUB is installed onto the HDD, even though the running system is only on the SSD.
Another easy way of checking if the HDD is running, is running something from the HDD, e.g. an OS. Use the hard drive for about 1 hour. Then note the temperature.
Now, let the laptop cool off completely.
Now, run something from the SSD, only. 1 hour later, check the temperature of the hard drive again.
It should ideally now be between 15 and 30 °C. If it is running, especially inside a laptop, it may get to 50 or even 60 °C (but it also depends on how heavily it is used).
You can also check if any process from your current SSD-only OS is accessing the hard drive.
Those are the three easiest ways of checking if the HDD is running. Perhaps, there are also better but more complicated ways to achieve that.