Parker elbowed Nashville's Jordin Tootoo late in the second period of his team's 3-2 win on Saturday. After leaving the ice, Parker tried climbing the glass barrier to get at Witt.
Neither Parker nor Witt, making his Nashville debut after a deadline trade from Washington, managed to land any big punches. Both were ejected after teammates and coaches quickly pulled them down.
Parker will miss games against Los Angeles on Monday and St. Louis on Thursday and will forfeit $5,971.42 in salary.