Speaking as someone who is going BACK to crating, I have to say crating your dog makes your life with them infinately easier. As long as your dog has a crate, he has a safe place to call his own. And you have the peace of mind knowing he's not soiling the houe, or chewing on things he shouldn't, or getting into chemicals or things he just shouldn't be bothering with.
Also what the other commenters have said is true - a chihuahua mix, or any small dog for that matter, shuold not be made to stay outdoors.
* They are prone to both heat and cold - where they can become ill fasther than a larger hardier dog.
* They can be carried away by other animals including but not limited to hawks, foxes, raccoons, dogs, cats etc.
* They are also animals that other unscrupulous people wouldn't hesitate to steal from your yard for the purposes of being sold to laboratories, as bait dogs to dog fighters, and possibly just resold for money to another family.
And chihuahuas by themselves have an issue with hypoglycemia where if they aren't fed often and well they can become sick due to low blood sugar. Keeping them outside is not a good idea for this very reason alone because you cannot properly monitor them or their eating habits.
If you can't get your parents to change their minds, I have to say, you may want to think about taking the chi-mix back to the humane society and get a more appropriate dog ... although no dog should be regulated to a life outdoors by themselves. They are pack animals as much as their wild canid brethren are... they need pack mates in order to be happy.
Good Luck!