We have two mini piebald daschunds who are normal for their breed--they love to sniff and eat whatever they find on the floor. My daughter wants a parakeet, and as I expect a lot of birdseed debris on the floor, I am wondering if eating this would be harmful to the dogs. Also if the bird got loose, I'm imagining dogs chasing a bird through the house... any input on these questions?

I am a staunch believer in feeding a dog dry dog food and not much else. As far as I know, there is nothing in bird seed that would hurt the dogs. I don't think there would be enough of it to upset their diets. Vacuum frequently.

As for the bird getting loose, likely as out of control as things might be, the bird could fly to a safe high perch. The dogs would be of little help in catching the bird. I once had a purple grackle get into my house. The first thing I did was throw the dog in the crate. If the bird gets loose, the first step would be to restrain the dogs. Open the door to put them out, and the bird might go too.

Parakeets make a very large mess around their cage - I had two for a while, so I can tell you from experience. They are great beginner birds, especially for kids, as they are quite friendly and trainable. The mess, though, consists of droppings, feathers, and food. There probably won't be enough food to upset the dogs' stomachs (although Dachshunds can have sensitive stomachs) if you vacuum frequently, but I'm more concerned about the feathers and droppings. These are more likely to upset their stomachs, and I believe dogs can get parasites and/or some diseases from bird droppings - I would ask around or research that first. I haven't done enough work with birds to know much about them.
Keep in mind, too, that birds stress very easily. If they aren't use to dogs and see your little ones staring at them, it can be dangerous to the birds. If you have a room or section of your house where your dogs don't go, it would probably be best to keep the birds there.
Good luck.

I have a dachshund and I got a parakeet at one time had to give it away, it flew by my hand as I was feeding it one day, and it flew around the house, would not lite because my dach was barking like crazy, it finally got tired and fell to the floor, then my dach saw her chance and grabbed it, I was screaming, and I finally got her to let loose, it was fine a little shook up though, needless to say I found the bird a good home dog free. As far as I know it is still doing well.