This weird-shaped cake pan is difficult to use when it comes to the step that I must line it with parchment paper. How should I do it properly so that the cake will easily be removed from the pan?
You can't effectively line a Bundt pan with paper. My favorite method is to mix cake release and keep it in the cabinet. It lasts for months and months. Just mix 1 part vegetable oil, 1 part shortening and 1 part flour (roughly, by volume). Brush that mixture in the pan, getting all the nooks and crannies. It doesn't make the mess that traditional flouring does, I always end up getting flour everywhere doing it that way. I think the mixed ahead release works better too. It also doesn't show up on the cake as unmoistened flour. With your non-stick pan plus the cake release, you'll have no trouble.
OXO makes a silicone pastry brush that's great for this and all kinds of tasks, I use mine all the time. It goes in the dishwasher, it's heat resistant, it doesn't shed. It's a very handy little item. Look for the strips of silicone in between the bristles with holes that hold on to sauce. You can see that in this picture using the magnification tool. OXO Pastry Brush