I don't think you're over reacting. Here are the symptoms of Cocci along with a description of what it is
http://ultimatefowl.atwiki.com/page/Coccidiosis. It definitely sounds like the early symptoms to me. Have you told your parents that if it is cocci all your chicks will die? Like I stated before, cocci is not something to mess around with. It is better to spend the money on the Corid and give them the dosage than have them all die and have to start over with a new batch of chicks. Make sure your parents understand that this is serious. Sounds like you are paying really good attention to your chicks and you know when one of them isn't quite right. That's the kind of attention you have to be paying to catch an illness when it comes to chickens. Like leadwolf said, chickens do not act sick unless they are really really sick. You need to convince your parents before it's too late.
Good luck!