Prepare the pineapple juice by placing the pineapple chunks in a blender and processing until smooth. Strain through a fine-mesh sieve to remove pulp, pressing gently to extract as much juice as possible.
Make a quick simple syrup by combining the sugar with 1 cup of water in a small saucepan. Heat over medium heat, stirring occasionally until the sugar completely dissolves (about 3-4 minutes). Remove from heat and let cool.
Juice the lemons until you have 1 cup of fresh lemon juice. Remove any seeds but keep some pulp for authentic flavor and texture.
Combine the pineapple juice, lemon juice, and cooled simple syrup in a large pitcher or punch bowl. Stir well to integrate all the flavors.
Add the honey and stir until completely dissolved into the mixture.
Pour in the remaining 5 cups of cold water and stir to combine thoroughly.
Gently muddle the mint leaves by crushing them lightly between your fingers to release their oils, then add them to the punch mixture.
Refrigerate for at least 1 hour to allow the flavors to meld together and the punch to chill thoroughly.
Just before serving, add the sparkling water and stir gently to preserve carbonation.
Garnish the pitcher or individual glasses with lemon slices, pineapple chunks, fresh mint sprigs, and plenty of ice.