In the bowl of a stand mixer fitted with a dough hook attachment, dissolve yeast in warm water. Let stand 5 minutes until foamy, then whisk in sugar and salt. With mixer running on medium speed, add flour 1 cup at a time until dough pulls away from sides of bowl, adding additional flour if needed.
Turn dough onto a floured surface and knead several times until smooth and elastic. Place in a greased bowl, turning once to grease top. Cover bowl with plastic wrap and let rise in a warm place, 1 hour or until doubled in size.
Punch risen dough down and divide into 12 equal pieces. Shape each dough portion into a 6-in breadstick shape and place 1-apart on two parchment paper-lined baking sheets. Cover with plastic wrap and let rise 15-30 minutes.
Make the garlic butter: In a small saucepan, melt butter over medium-high heat. Add garlic to pan and saute 1-2 minutes until garlic is fragrant. Remove from heat and stir in parsley and salt. Set aside.
Meanwhile, in a large saucepan, bring 6 cups water to a low boil over medium-high heat. Dissolve baking soda into water. Working 1-2 at a time, carefully place breadsticks in boiling water and boil 30 seconds on each side, flipping once. Transfer back to baking sheets and repeat boiling with remaining breadsticks.
Bake breadsticks in a preheated 450F oven 10 minutes, then remove from oven and brush with garlic butter and sprinkle with Parmesan. Return breadsticks to oven and continue baking an additional 5-10 minutes until breadsticks are a deep golden brown. Enjoy hot from the oven!