That's because, in Windows Forms, Transparent isn't. When you set the backcolor of a control to Transparent, what you are really doing is telling the control to take on the background properties of its container, likely your Form. You cannot see the Button control behind the Panel.
Opacity does NOT make a control transparent. It makes what you are drawing partially transparent, but it does not make the drawing surface of the control transparent. To get around this, you would have to make your own version of the Panel control so you get complete control over how the Panel is painted.
For an example, see
Creating a .NET Transparent Panel | CodeGuru[
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You have another problem. The Panel control you're going to create will cover the Button so any mouse clicks will go to the panel, not the button! The Panel is visually transparent to you, not the mouse.