From the Super Mario Wiki
Super Mario Bros. 3
Red Paragoombas make their first appearance in Super Mario Bros. 3 as a weaker form of Paragoombas. Their only method of attack is to hop along the ground, attempting to run into Mario or Luigi. To defeat them, the player must jump on them once to rid them of their wings and again to defeat them. Unlike their yellow counterparts, none of the levels show them wingless first.
Super Princess Peach
Red Paragoombas serve the same purpose in this game. A sub-species, called Mad Red Paragoomba, also appears. This species can cling to walls, allowing them to walk across high ledges without falling off the edge.
Super Mario Maker
Regular Paragoombas behave just like their red cousins from the upper-mentioned games in Super Mario Maker. Even though this behavior is remarkable, these Paragoombas are not of red colorition.
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