I just baked a 9×13×4 inch cake. I'm sure it is cooked through; 350°F for 50 minutes. The toothpick test came out clean, and when I presses on, it it was firm. 

Anyway, i let it sit on the counter for about 15 minutes still in the pan, and I just took it out and put in on a cooling rack where I'll let it cool for another 30min. The bottom and sides of the cake seem moist and almost "sticky." Could it be because I put butter on top of the parchment paper in the cake pan and on the sides of the pan to prevent sticking? And do you believe my cake is still fine? Thanks.

It is most likely from the butter.  But you don't need to butter your parchment paper.  The point of the parchment is to prevent the cake from sticking to your pan.  The paper won't stick to your cake so no need to butter it.

I also don't butter/grease the sides of my pans.  I cut a piece of parchment to fit the bottom and that is it.  Once the cake has come out of the oven I let it sit in the pan for about 10 minutes then take a flat metal spatula and run it between the cake and the pan to loosen the edges and flip it out.  Never have had a problem with stickage.

Oops, I hit submit before I was done!

I am sure your cake is fine.  The edges are just slightly "sticky" because of the butter you used.  But no harm done!

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I agree, probably just the butter.  I never grease my parchment paper either.  In fact, that is why I use it because I've had issues with buttercream not sticking to the cake when I used homemade cake release (equal parts flour, oil, and shortening).