While your parents won't purchase a copy of Minecraft for you, you can buy a Minecraft game card from a store (Gamestop, Best Buy, Target, Walmart, ect.) for $27 US for one copy, $49 US for a 2-pack (if you want to get a copy for yourself & a friend). The card has a code that you redeem on Mojang's website to upgrade your Minecraft account to premium (paid).
This is the best option as it doesn't require any credit cards, but you'd likely have to save your money / allowance to purchase said card. This option is probably harder for your parents to say no because they don't have to utilize a credit card… Or have others (like relatives or grandparents) purchase the game card for you as a present.
I Can't RECOMMEND ATTEMPT TO GET A PIRATED COPY OF THE GAME! While Mojang has a relaxed stance on piracy, pirated copies can potentially contain unwanted "extras" (viruses, malware, & other undesirable stuff) bundled with it that might get you into trouble with your parents in more than one way.
The Demo version of Minecraft does allow you to enjoy each world you create for 100 minutes before locking down into read-only mode (which gets unlocked when you pay for the full version), but it does allow you to enjoy LAN play (which is an unintentional thing, but the demo version can't control the world that's operated by the other player(s. The full version removes restrictions from the demo version & allows online play as well as custom skins on your character.
As for other games to consider… I'm sorry that I won't be the best assist since I haven't scrounged around for free games. However, if you do get into Steam, games similar to Minecraft to check out would be…
* Terraria -- Like a 2D version of Minecraft, but has more action / adventure elements.
* Starbound -- Pretty much "Terraria in Space" (each planet would be consider a separate world within Terraria, but all stringed together with a Sci-Fi twist).
Both of those games are paid games as well, but are available on Steam & DRM Free.