TLDR: Look up a part via the Parts/Search or by navigating to a part view through a BOM or other page. Select the Shopping List you want to add the part to and the multiple of the part you want to add to the Shopping List. Click the "Add" button and the part at the appropriate multiple will be added to the Shopping List.
All the details:
To add individual parts to a shopping list from a parts page, just click on the "Add To Shopping List" button in the upper right corner of the parts page.
Once you click on the "Add To Shopping List Button" you'll see a modal pop up asking you to pick a shopping list and enter the quantity of the part you'd like to add to the Shopping List. The Shopping List selected will also show you the shipping address and number of lines in the Shopping List.
I'll choose "Fall Production" and add 50 of the JS102011SAQN to the Shopping List which looks like this:
Click "Add" to add them to the Shopping List.
If you navigate to the Purchasing app and then Shopping Lists, you'll see the Shopping List now has a larger number of items. For me, a total of 1 item. If you click on the Shopping List it will open and look similar to a BOM, but with a slightly different interface, like this: