I am playing Icewind Dale: Enhanced Edition, and am at the top level of Dragon's Eye. My party is as follows

Fighter x 2
Mage x 2
Thief x 1
Cleric x 1


The game is great, I am really having a great time... But I am also interested in trying some other classes too. Here's the question that I have (and I hope it doesn't sound dumb) - should I start a new party with new classes, get to where I am with my current party, and then play them parallel to each other, or just play the party I have now to the end and then start a new one? I am curious as to everyone's take on this.

Can someone recommend a few classes too?

I'd be inclined to say "Play one party at a time". I don't see much reason to be doing the same events in parallel in two different saves with two different parties. Just focus on one play-through at a time; you can always start over and do the whole thing again immediately if you are so inclined.

You have a good mix of classes. If you want some character variety, then try taking some of the kits for each class, if you keep the characters single-class. Alternatively, multi-class some of the characters to expand their list of abilities or change their style of play. For example, a fighter/mage is going to function differently from a berserker (one of the fighter kits).

Edit: And a Thief/Mage is going to function wildly differently from a Fighter/Thief.

Great! Thanks so much for your advice and help!