I would be very cautious when buying stuff related to battery charging on Amazon. In my opinion, Amazon is great for buying regular consumer products (Food, Games, Clothes, etc), but I would be skeptical when buying low level electronic equipment. To put it simply, Amazon doesn't have the best quality control in this department, so you may end up buying something that doesn't work. To add, the parallel plate you linked to seems to be prone to failure in some cases based on the reviews. That alone would be a turn off for me since you don't want a Lithium Ion/Polymer related product to fail.
That said, parallel charging is possible. You just need to make sure you do the following: 1) Make sure you are using good battery cells and not cheap knockoff cells. 2) Make sure the batteries you're combining in parallel have the same voltage level (probably no more than a 100mV difference). 3) Understand what the charging current is per individual cell. 4) Use the exact same cells when charging in parallel. Whatever the recommended/max charging current is, multiply it by x number of cells in parallel. As long as your power supply has enough power to charge your batteries, you should be fine.
Also, if the charge current is fixed, would the charge time proportionally divide by the number of batteries I have?
The charging time will be about the same if you scale the current up, but will take significaltly longer if you have multiple batteries in parallel. But yes, you're understanding of charging time is correct.