If you can do the internal swap, I would suggest doing that. I am sure it would be a huge improvement.
But, if you don't trust yourself opening the iMac, your external options for that year iMac is USB 2.1, TB1, and FW800.
If you can find a cheap TB drive for a SSD, this would be recommended, as the USB and FW would be around the same read/write speeds as a HDD.
On eBay, I bought a few used LaCie Rugged drives, the one with TB and USB3.0. They had HDDs in them, and I swapped them for SSD, it is super easy.
Also, upgrading the RAM is super easy, and would probably improve things. There are four slots, and probably only two are being used. You can install up to 32GB of RAM, so if the two other slots are free, get two 8GB sticks and that would improve things a lot.
I would look into doing the internal replacement. There are plenty of how-tos, any is relatively easy.
Use suction cups, pull the glass, remove 8? screws, pull up on the display.....
While I would totally remove the display to prevent pulling on the cables, you could probably do it just by lifting it up with a second person.
Unscrew the mounting screw for the HDD, put in new SDD with 3.5" adapter or just tape it down.
I think you will need an external temp sensor for the SATA cable. I did that for a 27" that I swapped the HDD out for a SSD. It was $35 on Amazon. You could also use software solutions, or I have seen people just cut those cables and short them. I would just go with the sensor.
That reverse to put everything back together.
Enable TRIM.
do you think that would correct the problem and make the iMac faster
Before making too many investments, maybe do some trouble shooting to make sure that the problem is your HDD. By your symptoms, it sounds like a failing HDD.
Try using Disk Utility, and scan the disk to see if it needs to be repaired.
You can do Apple HW diagnostic, but this often has false negatives, meaning your HW is failing, but the diagnostic says it is fine.
You can just do a quick external solution for trouble shooting, maybe get a cheap USB to SATA adapter and a cheaper SSD, both you can get for less than $30. Or you can use a thumb drive, but it will be slower. Install MacOS on the external drive and see how everything runs. If everything seems okay, then your HDD is probably failing.
Is this your only Mac? If not, you can use your other Mac for some trouble shooting too.
or I am better of buying a new iMac?
I am a fan of using old Macs if they get the job done without any trouble. If it is as simple as a failing HDD, and you parents are otherwise happy with the iMac, then I would just do a SSD.
Don't get an expensive new Mac when a older one works fine.