Preheat your oven to 325°F. Line a 9x9-inch pan with parchment paper and set aside.
In a medium bowl, sift together flour, baking powder, baking soda and salt; set aside.
In a small bowl, mix together yogurt, cocoa powder, vinegar and food colouring.
In a medium bowl, beat butter with the vanilla using an electric hand mixer on medium-high speed until creamy and whipped, about 30 seconds. Add sugar and beat until light and fluffy, about 3 minutes on high. Add egg and beat until well incorporated and fluffy, about 30 seconds on medium-high speed. Mix in oil.
Add half of the flour mixture and beat on medium-low speed until mostly incorporated. Add half of the milk and beat until well blended, about 10-15 seconds. Beat in the remaining flour mixture followed by the last of the milk mixture on medium speed just until the batter is smooth, but do not over beat. Set aside ¾ cup of the batter and fold the red yogurt mixture into the main batter.
Dollop red velvet batter alternating with vanilla batter into pan and swirl it in to make a marble effect. Bake 25-30 minutes until a toothpick inserted into the center comes out clean. Transfer to a wire rack to cool completely.
For the frosting, beat soft butter with icing sugar until smooth and fluffy. Beat in soft cream cheese one tablespoon at a time until full incorporated, thick, fluffy and pale in colour. Beat in lemon juice and vanilla until the frosting is smooth, fluffy and very creamy. Spread it over the cooled cake and decorate with sprinkles!