Update:

I got the thermostat from Whirlpool. I installed it and finished the
project. Everything looked just like the picture and I read the
directions several times making sure I did not miss anything. At this
point I was very confident and feeling a little proud of myself. I
light the pilot and I see the light through the window. Then I let go
of the red button and the pilot light goes out. I now have replaced
every gas part on this water heater and I still have the same
problem. I give up.
I called my home owners warrenty company that was bought by the
previous owner of the house when I purchased it. I regret not calling
them first. I called Whirlpool first thinking they would stick by
their product. I slipped up on one phone call and mentioned I was the
second home owner and after that they told me the warrenty was void
and they did not want to work with me after that.