I really feel for you, Goat. I am constantly aware of how little respect we show young people. I remember being thoroughly frustrated by it as a teenager and I still notice and regret it. It's true that teenagers can be SOOO annoying (come on, admit it, you probably think so too, even though you are one! LOL), but it's just wrong to expect them to behave like adults but not give them any of the privileges or respect we show adults. (And just so you know I'm not prejudiced, I think 36-year-olds can be thoroughly annoying too. And 80-year-olds. And 25-year-olds. Annoyingness doesn't discriminate based on age...)
ANYWAY... given that I can't change the laws or the way people around you treat you, all I can do is offer sympathy and a few thoughts on things you might do to help the situation. First of all, it seems clear to me that the evidence points to the neighbor's dogs in a way that is pretty much beyond reasonable doubt (physical evidence of violence in the nest, physical evidence that the culprit was canine, and then the circumstantial evidence of the body found in the suspect's driveway--it's not rock solid, but in a human murder case I suspect it would be enough to press charges & start a forensics investigation). I guess that if you had witnesses & photos of the aftermath, a reasonable jury in a court of law would convict your neighbor's dogs.
But that is really a moot point right now, because without the photos and the support of your parents, there is not much you can do about it. And anyway, it may be that "proving" and "pressing charges" may not be the best approach for accomplishing what you want.
I've been taking some conflict resolution training, and here is an approach you could try that would work despite your age and will probably lead to better relationships for everyone involved. Start by thinking about your neighbors, and specifically trying to understand how they must feel right now. Here is my guess: They probably feel pretty bad about what their dogs did, even if they aren't ready to admit it. People are funny sometimes--when they feel worst and guiltiest about things is often when they are least ready and able to say "sorry" or admit the guilt. They are probably feeling defensive and on their guard, because they know you're pretty mad. When people feel defensive, they try to find ways to blame someone else: "The ducks were in our yard, what do they expect?" "Who keeps ducks as pets anyway? They're just prey animals asking for trouble!" "Why do they have to be so emotional about it? Life happens!"
But behind those defensive statements/feelings, are people like you and me who just want to feel right with the world, and who really would prefer to have a good relationship and to feel right again, if they only knew how to get there without making themselves vulnerable.
Once you understand that, you can talk to your mom and then go to the neighbors with a formula that is designed to take people off their guard and promote better relationships and happier outcomes to conflict:
The formula goes like this: Observation, feelings, needs, requests. Here's how it works:
Observation, which in this case should include an effort to take them off their defense by acknowledging how they probably feel about the whole thing, and giving them credit for the good things they do for/with their dogs: "I think you must feel pretty bad about what happened to my ducks. We all know the dogs were just doing what is in their nature to do, and I'm sure you do your best to keep the dogs contained."
Next, talk about your feelings (without placing blame): "But I admit I'm pretty torn up about it. I really love my ducks and am so sad about what happened."
And then your needs: "I need to feel like my ducks will be safe from now on."
Finally, make your request: "Can we talk about what you think we can do to prevent the dogs from hurting the ducks again?"
You can use the same formula to ask for compensation, if you and your mom decide that's best. Here's what it would look like:
Observation: "I lost 5 ducks today."
Feelings: "I'm sad and frustrated because I can't afford to replace them right now, especially when I also have to spend money to increase their protection."
Needs: "I would really like to add some new fencing and probably order some new ducklings for the Spring."
Request: "Would you be willing to help me pay for some of this?"
If you get a positive response from that, be ready with some reasonable numbers to discuss. Don't be set on a specific total, though--leave some room for them to make suggestions and to offer some generosity, so they feel like they are a part of the process, and not just the recipients of your penalty.
The compensation discussion may come naturally out of the first part--while discussing how to prevent the dogs from hurting the ducks, it may come up that you are going to some expense, and you can take the formula I just described (observation, feelings, needs, requests) to start the compensation discussion.
By the way, you can use the same formula to talk to your dad about how you feel about his attitude toward the event: "Dad, I noticed you telling the neighbors that this is "just life" (observation). I felt sort of sad and like you weren't really sticking up for me (feelings). I want to know that you stand behind me when things like this happen (needs). Next time, could you talk to me first so we can decide what we want to say to the neighbors together (request)?" In fact, you can use it any time you talk to anyone when there is some sort of conflict present or expected.
I have found it to be a really powerful way to interact with people. It takes time and practice, and I am far from perfect in my use of it. I stumble and sometimes forget to use it at all. But I'm hopeful that as I practice it more and more, I'll get more and more benefit from it.
Whatever happens, I hope you find a resolution that feels right to you.
Edited to add: You probably already realize this, but just in case, I wanted to clarify that the sample dialogs I wrote are just samples. You'll fill in your own observations, feelings, needs, and requests, according to what's in your own heart.