Lovely Pena (or Puna) Lhakhang is ignored by most visitors to Paro, but is said to have been founded by Tibetan King Songtsen Gampo in the 7th century, making it one of the oldest temples in Bhutan. It's easily missed, just past Dumtse Lhakhang, on the east side of the road.
The main inner sanctum certainly has an ancient feel, dominated by a statue of Jowo Nampar Namse that apparently has the power to fulfil wishes. The red-faced protector Pehar lurks in the corner, while to the left of the chapel is the stone footprint of a former Zhabdrung. You may have to call the caretaker to open the building.