I think the people who write these headlines and articles need to be smacked around a bit for making them so misleading, and also I think people actually should read the article before responding to the title alone.
The document referred to in the article doesn't actually say anything about "giving planets the same rights as humans" (the article does, but that's bad reporting for you), it just suggests that the Earth may have "rights". it's essentially suggesting that humans need to temper their exploitation of the planet and realise that there are limits that need to be adhered to if resources are to be sustainable.
They may phrase it as "rights", but in a way I think it's about giving Earth a "voice" to say "OK stop now, that's enough". Whether you think that appraoch is reasonable or not is up to you, but it's not something so ridiculous as giving the planet "human rights".