Assign a Pokemon with the Crush specialty to produce Pink Paint passively. You'll still find it while exploring, but the helper output adds up. It goes into 3 recipes, so having a passive supply helps.

Paint used to color items pink

Raw Material
Pecha BerryCrushCrushPink Paint
ShopSmearguru / Palette Town Shop
Bleak Beach (Smearguru hut), Palette Town (Level 5)
GeodudeCrushAurorusCrushShieldonCrushElectivireCrushCranidosCrushOnixCrushSteelixCrushPupitarCrush

There are 3 recipes that call for Pink Paint, and 3 of those are Other items. Scroll through or filter by category to find what you're looking for.

3 recipes
Paint Balloon PinkOther
Pink Paintx2
Paint Balloon PurpleOther
Pink Paintx1Blue Paintx1›
Paint Balloon RoseOther
Red Paintx1›Pink Paintx1