He was ironing towels. Even penguins need towels.
Pingu’s dad is a complex man (penguin).
Did any of us truly know him? Can we confidently state his motives for doing a lot of ironing?
Perhaps there was something about the repetitive activity that he found deeply soothing.
Perhaps he is an ex-military man (penguin) obsessed with imposing order on the chaos of the world in any small way he could find, and ironing creases out of towels provides him with some small solace.
Perhaps by doing the ironing he could successfully argue that he shouldn’t have to do the washing up as well, the chore he most hates.
We can’t be sure, but in the literally minutes of research I did, I could only find footage of him ironing what appear to be towels, or possibly sheets.
And while his penguin family don’t wear clothes, they do have the kinds of domestic arrangements that mean bathing and sleeping in beds is not inconsistent with the logic of the Pingu world.