You basically cannot use で with these movement verbs. A rare exception is when the action does not involve the actual change of place, or when such a change of place is not important at all as the purpose of the action. (EDIT: Another exception is when で is clearly used as a method marker, e.g., 電車ではなく高速道路で行きます "I'll go not by train but using a highway".)
この道で歩く is almost always nonsense, and you should say この道を歩く instead. However, if the context is a strange YouTube streamer who sets up and uses his treadmill in different places, then saying 昨日はこの道で歩いた, 今日はトイレで歩く, 明日は図書館で歩く and so on make sense because he stays in the same place while doing the action of walking.
公園を走る is much more common, but 公園で走る is also acceptable if the context is where you do your everyday exercise (the purpose is the running itself rather than the change of place). If the park is relatively small and you run over the same place many times, 公園で走る can be more acceptable. Likewise, プールで泳ぐ is typically more natural than プールを泳ぐ because the purpose is usually recreation or exercise rather than traveling (you move back and forth within the same narrow area).
鳥が空で飛ぶ usually makes little sense, but it can be used when talking about a wounded bird that has been training to fly in a facility. When it finally tries to fly in the sky as the last step of its training, you could say 明日その鳥は初めて空で飛ぶ or something like that (the flying action and the location are important but the traveling is unimportant). 僕は部屋で飛ぶ is a reasonable catchphrase of a flight simulation game (you are experiencing the flying without actually traveling), and in this case 僕は部屋を飛ぶ makes no sense.