Question: Does anybody know if there is a website where I can record my incomplete LEGO sets?

That way I could generate a list of parts I would require in order to complete them.

I currently use Brickset, however that only allows the user to record completed sets.

Answer: 
Rebrickable.com offers this service. First add the complete set to you list, then view it on the set page like this <http://rebrickable.com/sets/42029-1/customized-pick-up-truck-technic-2014> At the bottom of the first panel you'll have an option to "Add to my parts list" from then you go through the list of parts and remove the ones you are missing.

[![enter image description here](https://i.stack.imgur.com/wwZkd.png)](https://i.stack.imgur.com/wwZkd.png)
Not a website per se, but look at [BrickStock](http://brickstock.patrickbrans.com/). This software runs on PC, Mac and Linux (via WINE) and is free to use (Registration and payment is only needed to use the Seller features of the tool). It allows you to download set inventories from BrickLink and creation your own inventories.

Here is how I would use BrickStock in your scenario:

1. Download the inventory for the set you are looking at.
2. Get the last 6 month average sale prices (Ctrl-A Ctrl-G) to help weeding out alternates (See step 3)
3. After removing CounterParts (marked as CP) and alternates (marked with identical numbers, e.g. 1 and 1, 2 and 2 etc., remove one of the two entries (typically remove the less common part i.e. the one with the higher selling price)), and optionally all the parts marked as "Extra", set all parts to Status "Exclude".
4. Then, go through your incomplete set inventory and for every part you have, set the entry in BrickStock back to "Include".
5. When done, you have a list of "Exclude" items which are the pieces you still need. Alternatively you can change the logic and make the default "Include" and mark all the parts you have as "Exclude".

You can save Inventories to file, and have multiple set inventories open at the same time.