Some say that the difference is that blanched food is thereafter given an ice bath to stop the cooking, a step not necessary when parboiling.
But that’s not really accurate.
For example, nuts are blanched to soften the skins to make them easier to peel off but they aren’t given an ice bath after blanching. So, it may be more accurate to say that nuts are parboiled rather than blanched to soften the skins to make them easier to remove. In that context, perhaps, the difference between blanching and parboiling is simply a matter of term usage.
For me, the main difference is whether, after the process of boiling, the food is fully or partially cooked. If the food can be eaten after boiling, with or without the ice bath, it was blanched. If the food needs additional cooking, it was parboiled.
How to blanch
Start with plenty of boiling water — plenty enough to submerge the vegetables in. I like to add salt but that’s optional.
Plunge the vegetables in the boiling water. Push them down during the first few seconds to make sure that every part gets in contact with the boiling water all at once.
And here’s something worth remembering: When blanching vegetables, make sure that every part of the veggies will cook at approximately the same time. If the vegetables have thick tough stalks and leaves, it may be best to cut them, separate the tougher portions of the stalks from the leaves, then blanch the stalks longer than the leaves.
How long vegetables should be blanched depends on what vegetables you are blanching and how soft you want them. For instance, pechay (bok choy) or something as delicate will be done in about a minute. Chinese broccoli (in the photos above) takes a few minutes longer. The length of blanching time needed by each kind of vegetable is something you will learn over time just as you will realize that more mature leaves take longer to cook than very young sprouts.
When the vegetables are done to your liking, scoop them out at once. A kitchen spider is really useful for this job.
Plunge the drained vegetables in iced water (which you should have prepared before or during the blanching process). Or place them in a colander and let cold running tap water cool them. Then drain the vegetables.
Depending on the kind of vegetables you blanched and how you intend to serve them, you may want to squeeze out the excess water. You can do this by lightly wringing them or by placing them between kitchen towels and pressing down lightly.
How to parboil
When parboiling, food is boiled in water (or some other liquid), removed, returned to the pan and cooked with the rest of the ingredients for the dish.
Why does food need to be parboiled? Different reasons. One reason is to tighten the skin of a bird such as chicken or duck. Another reason is to remove impurities.

For example, in Asian cooking, when braising or stewing meat, the meat is parboiled to allow scum to be expelled into the water. The partially cooked meat is rinsed not really to cool them down to stop cooking in residual heat but to make sure that any trace of scum is removed. The meat is then drained before going back into the pot.