I've noticed that lots of people are having trouble with their slim tools. I was one of them. Here are very clear instructions on how the pieces fit in.
Piece A: did not fit at all for me. Roll up a ball of toilet paper or tissue, stick it in. Make sure it's big enough to not get sucked down into the depths of the PS2.
Piece B: Use a screwdriver to move the disk brake aside. Just stick the piece in there, with the "dent" facing the hole. The instructions from the link I provided work well.
Piece C: The hardest one. First take a nail file and file off the little bump on the "ridge" of the tool: Now the tool will fit. Position it directly over the hinge, if it does not go in immediately follow these instructions:
put your finger on the top, take the small screwdriver in your other hand and tap it left, right, forward, or back depending on which edge you think is not letting the tool go down. Press pretty hard and it should slide in about halfway.
Now you need to push it in untill it is flush with the bottom of the PS2 case. You may have to pull it towards you a little and push it in at an angle. Try different stuff, as this is the hardest part of the slim tool installation. Try wiggling the tool around until you find the "hole". Try pulling it up and pushing it back down. Use the screwdriver to help you push on it. If you do get it in, but it's at an angle, as in the back of the tool sticks up a little, push on it with the screwdriver and it should snap into place. You're done!
Problematic points-
If tool B won't stay in, hold it in position with your finger while holding the disk brake out of the way. Let go of the disk brake when you have a comfortable grip on tool B, grab your screwdriver and push on the bottom of the tool (the little ridge). That should put it in firmly.
If you haven't already, file down tool C according to my instructions! I can't stress this enough! If it still doesn't fit, make sure that it is filed down enough. Sometimes little peices of plastic can be "shaved up" prevent you from pushing it in.
I hope this helped. The slim tool is the best way to go for playing import games, as the tape mod cost me my console (luckily I was able to return it for a full refund and get a new one :P ). Have fun playing your imports!