Free museums ? In London yes, but not around where I live. There are big train and parking transport costs to go to London.
There are small parks that are walking distance that are fine around here, a few teenagers hang about, but yes we go to them. But that's not all day is it ? Maybe 20 minutes, longer if he has a friend to play. Some bigger parks are a drive away and you need to pay parking or pay entry to some. One nearby is £13.50 for age 3 +, we have never been to it. If you cannot even imagine not being able to afford £26 for a day out then you are very lucky.
Btw for my birthday I don't get anything as we don't do presents due to finances, although this year I did treat myself to a pack of new socks from Tesco's due to mine being like nets. And I don't live in a one bed flat although a flat can be massive,---- but yes my garden is pretty small, like most of the properties in my area as it's a newish town.
I could work, but I wouldn't be cherishing the holidays then as my child would be at a childminders (as well as before and after school.) Unless of course I work in a school or get a job with very an extremely generous holiday package. My DH can't help with holidays, or school drops, as he doesn't get paid if he doesn't work so only takes the absolute minimum. He works 10 hour days plus travel. I can't use family. My siblings do not have kids.
And I adore my child, he is however very hard work.