Where i live day cares are free - only Problem is that there zero spots available. Maybe try again in 2022.
Yes, artificially fixing prices below the market clearing price inevitably leads to shortages. If the government wants to ease the financial burden on parents then a better solution would be to just give them money and let them pay for day care on the open market.
Where I live the cost is dependent on the parents' income, and where I come from it's a fixed cost that is low enough for most people to be able to afford it. Even if they lived in my country, I don't think I would call my parents to take care of my children very often.