Landlords can discriminate against pet owners because of insurance claims. You need to learn that landlords / property owners are required to have insurance and insurance companies write policies that prohibit any claims against animals. There are insurance policies that will accept pets but those policies cost more money. Therefore it's cheaper and easier for landlords not to accept pets.
You need to pay for the pet insurance premiums and show the insurance policy to landlords. Once they see you have pet insurance, any landlord will rent to you.