Do children need to apply for a visa extension in Thailand if the adults do?

Legally all kids need an extension otherwise they're on overstay. Many people don't bother because kids can't be fined, deported or banned. However, they will get an overstay stamp in their passports, which for Thailand doesn't mean a thing. However, when travelling to other countries, an overstay stamp can (I'm not saying will) raise a red flag. So there's your options

Sounds amazing! Are you coming on a train?

You might also want to be aware that there is , apparently a new tourist fee of 300 baht starting 1 June .

If you arrive before April 1st you will get 45 days visa exempt as long as your passport is from a visa exempt country. After April 1st it's 30 days on arrival.

Tourist extensions are for 30 days from the local immigration office.

I don't remember the age but it might 14 or 15, children under that age cannot be fined or banned for overstaying, so if they are young children and you don't purchase an extension for them, they aren't likely to have any issues in Thailand.

An overstay stamp in the passport might cause an issue in some countries though, so if you're traveling a lot you should take that into consideration.