Place peaches in a blender or food processor; process until smooth. Press peach puree through a fine sieve into a bowl; discard solids.
Combine rind, lime juice, sugar, and mint in a large pitcher; crush juice mixture with the back of a long spoon. Add peach puree and rum to pitcher, stirring until sugar dissolves. Stir in club soda. Serve over crushed ice. Garnish with mint sprigs, if desired.
Use a wooden muddler to crush the mint mixture in the pitcher, if available. The procedure releases the mint's essential oils, melding them with the lime juice.