Sibling rivalry is a natural part of growing up; there's no way to avoid it.  Your best efforts won't avoid it.  We can help minimize it, and sometimes redirect it but there's no way to avoid it.

Sibling rivalry is probably at its worst when kids are all under the age of 4.  When they're less than 3 years apart, they're very dependent.  Think about it:  they can't go cook a microwave dinner.  They're very dependent upon "Mommy," so subdividing "Mommy" is a threat.
 
If you look at dogs, the females have what looks like a million teats, so if they have a large number of puppies born, all the puppies get to eat.  If you have 3 kids, you don't have  3 breasts, which makes it a little tougher and you usually don't breastfeed two at the same time.  I don't know why not, but you don't.  So as far as resources of love and attention go, you've got one "Mommy," and many demands.
 
From age 4 and up - competition between brothers and sisters can heat up.  It's usually the worst between 8 and 12 because if they don't have the same interests, there's pull and push and who's better, who's smarter, who is more important, who gets more attention, who does better in school, who does better in sports...all of that.
 
So here are some quickie ways to handle conflict between kids:

You have to treat each kid as an individual. Parents tend to fall into the trap of "I'm going to love and treat all my children the same".  Well, the kids are not the same - they don't have the same personalities, they don't have the same needs, they don't have the same emotional reflexes (much less physical reflexes).  What parents should focus on is identifying and reinforcing the diversity of talent:  i.e., "you're unique at this and you're unique at that."  And it's really good to sit with kids when the younger ones are looking, for example, at the amazing talent of the older one or sometimes it's the other way around.  And you sit down and you go, "Here's the deal.  You have Mommy and Daddy here.  Mommy is very good at 'blank'; Daddy is not so good at 'blank'.  Daddy is good at 'that' and Mommy's not so good at 'that' because we're different people.  And when Daddy does really good at 'that', I applaud.  And when Daddy sees I'm good at 'that', he applauds.  So we're happy about the fact that we're different and we have these good things to applaud."  And you teach your kids to do the same thing.  "You are definitely fabulous at math, but you are also incredible at art.  So when your brother or sister needs to do an art project, you ought to help."  "When you're having some trouble with math, go to your brother or sister.  They'll help you."

Having a sibling in the position of administering parental support breeds a bond as long as it's not done as a discount.  Any time kids are getting along try saying: "That's great how you guys are playing.  I really like seeing that; it makes me feel good.  You both look so happy and, you know, you're working things out.  That's really nice."  The more you can look for the times that work and make a comment, the better.
 
You've got to really spend time with each kid alone.  Everything can't be a team effort.  There has to be special time where you go to the library with one, a ball game with the other, a museum with this one, lunch with that one...they all have to have special time...reading, taking a walk, running an errand...special time. 
 
Look at how YOU are getting along with your spouse.  Poop rolls downhill (unless it's stuck in something).  So when you're bickering with each other with the criticism and the anger and not being happy, the kids will do it with each other.  The tension works that way.  Through words and actions, you've got to be very love-ish:  a lot of hugging, a lot of kissing, a lot of tweaking, a lot of cuteness...just a lot of cuteness.  I mean, my kid is 25, 6'2", 208 pounds and when I see him, I come behind him and I give him a big smooch on the top of his head and mess his hair.  Of course, if you mess your kid's hair, you're going to get in trouble. But, short of that, always be very affectionate.  It's a very important part of life.

I really think parents who try to get their kids to always do stuff together are making a mistake.  Kids need their own time, alone time and their own friend time.  So you don't tell your kid, "Bring along your younger sister or brother."  Don't do that.  Don't ever do that.  They need their own time with their own buddies.  If you want a babysitter, pay them 5 bucks an hour.  It's very important to have kids feel special and you can rotate: special kid of the day.  Okay, we do this in this order: 1, 2, 3, 4...(however many kids you have)...in that order, you're the special kid and you get these perks (and we have a list of perks), like you get to choose the TV program at 7 o'clock.  And the next night the other one gets to do it.  It doesn't matter how old anybody is -- they all get the "special kid" treatment so they're not fighting over a TV show because tonight that one gets to choose.  Of course some of you are nuts and have a television in every kid's room and I want to pinch your heads off.
 
Some of the things I don't want you to do:

Don't compare one kid to another. "Well your brother/your sister doesn't 'blah blah blah'."  Don't do that.  "He/she studies; you're just a bum..."  Don't do that because they'll hate each other. 
 
Try not to take sides.  Try not to take sides when they're having a little skirmish.  "Well you said...well you did...and you did...and you pushed..."  And say, "Well you know what?  At this point, I don't care who started it, you're both finishing it.  That's it.  If I hear more noise about this, you both don't go out for the whole weekend.  It doesn't matter whose fault it is, the two of you have to finish it."  It's the finishing done well that I'm interested in so they have to become a team or they both get screwed on the weekends.
 
Don't over-react.  You really shouldn't discount emotions.
 
"I hate Johnny.  I hate Mary."
 
"Okay, why do you hate them?"
 
"Because they do 'such and such'."
 
"Well I can understand how you can get an emotion so big you'd say 'I hate them', but you can't take their stuff or bounce them over the head or call them bad names.  When you feel a feeling, you feel the feeling and we can talk about what to do with the feeling, but these are the things you're not permitted to do with the feeling: you can't hit them, you can't take anything and you can't embarrass them, and you can't do crap like that.  But if you're that angry, you're that angry.  So you can either come to Mom and Dad and talk about why you're so angry, talk to your brother or sister and tell them you're angry, we can sit all of us and talk about why each one of us is angry because angry happens."
 
You don't discount the emotion because it's bigger.  In order to get accepted, it gets bigger.  So you say, "Oh, I can understand why you were angry.  However, bopping them on the head is not the way you're going to handle angry.  It's unacceptable.  But, you're angry, so if that's what happened, I can understand you being angry.  I would be angry too."  The minute you say that, the anger level goes down.  The minute you justify the anger, the anger level goes down.
 
So I could go on for days, but these are some basic tools that you can try and they all require you to have a sense of humor and be calm. Have a sense of humor and be calm because the more you get into it, you exacerbate it.  And definitely do not have parents arguing about it.