We fly SWA and this problem goes away
Exactly!!!!
As I mentioned on the thread regarding paying for EBCI on SouthWest airline (WN), just be THANKFUL it isn't a legacy airline! Why? because we have this exact situation on everyone but WN.
My suggestion to the OP up front (before it gets lost in the minutia below

:
If you can, find a middle seat and book yourself (and DH if going) into it. Next, book your little ones into aisle seats (preferable) and window seats (next best) if you can't get together. In addition if you can afford to put 1 or two into Economy+ via a fee, do this. These are trading credits you can use if required while on the plane. Most guys traveling alone will be happy to trade one aisle seat for another, since it is a straight swap. In addition, an aisle or window E+ is very trade-able for someone in straight economy.
Onto the rant

:
What exactly are Airlines thinking when they rebook you on a different flight and separate a parent and child? Do they just know that it will be taken care of on the plane?
This is a great question and definitely one we see
A LOT on here

.
As previously answered, and answered many times before, the airline (an incorporeal entity) doesn't think. It is a computer that does the assignments, not a person. I am honestly quite surprised that people expect humans to be doing this task (based on the content of the first sentence in the OP) or that computer systems work well enough to figure this out (the programing is quite sophisticated but still not nuanced enough to determine who is flying with whom most of the time).
Lesson learned: when people get involved, the concerns can be fixed; however until flight time, people are not willing to override the system.
This happens often with all airlines when they change flights- with internet fares the computer has no way of knowing there is a child/children involved. Most people don't know this but no airline guarantees seat assignments and even when you pay for a preferred seat you could be moved and your $ refunded.
While the computer systems absolutely know and understand the ages of the passengers, it isn't important. Children frequently fly unaccompanied by an adult, and the airline has rules regarding this. As such, it would be prohibitive (both in cost and functionality/maintenance) for the system to single out each passenger based on age, determine if they have a companion, and attempt to keep these people together.
The OP may not realize that these reassignments happen quickly to ensure those who are impacted by irregular operations (weather, mechanical issues, etc) are re-routed quickly and efficiently.
My suggestions-
Keep watching to see if seats open up prior to the flight and select them online or call asap. If this doesn't happen;
Arrive at the airport early and explain the situation to the agent checking your bags. Remember that he/she didn't cause this and enlist their help - the more understanding you are the better! Be willing to switch a good seat for a poor seat.
Ask again at the gate - again remember that the gate agent didn't cause this either but may have some options that will get you seats together.
If you are still separated by the time you board you can ask nicely - offer the less desirable seat for the more desirable seat and hope for the best. If a child is school age the FA's will not do as much to help you as they will for a younger child.
ABSOLUTELY!!!

Great points all around.
The ticket agent will be able to correct this at the airport, but you'll need to be there before anyone else that needs correcting. The 2 hours prior to flight is a great marker, as you will be there in time to ensure you have the best chance of getting seats together.
However as I stated in the opening monologue, if you are already separated, try to select some choice seats, as they make great bargaining tools should option 3 come into play.
So you are telling me that my 2 years old could be in a seat alone 5 rows from me?
It's totaly ridiculous!

This is a funny statement to me, since the outrage is completely misplaced and
provided without the proper context of the rules of the game. It sounds like it is simply reactive in nature and provides no meaningful assistance to anyone.
For your benefit, American Airlines (AA) Conditions of Carriage are posted below. They are generally the same as almost every other airline on earth. As such, it may be helpful to get a basic understanding of the rules before providing definitive statements of derision.
AA's
Conditions of Carriage.
Most appropriate to the OP:
In the end, almost ALL United States based airlines will play the same "seat swap" game with you (except WN, as I said those flying WN should thank God as they only need to worry about boarding sections). It is a matter of fact, and everyone reading this needs to internalize this information.
Even those with significant elite status are not immune, so those of us who fly infrequently and show little to no allegiance to a single airline need not complain.
READ the CoC of one of the major airlines to get a good feel for the rules!!!! Knowledge is power, and if you know the rules better than the person next to you, you will know how to get help from the airline staff, while the ignorant will only be able to scream or cry. If we fly once per year, looking at the CoC (at least) once is a good idea.
This isn't meant to be harsh; rather, it should act as a wake-up call. While I appreciate a good vent post, unfortunately, some post disparaging remarks against the airlines (in many threads) without understanding the game. While I completely agree that once I have chosen my seats a computer should not be allowed to reassign them arbitrarily to the benefit of those who book later than me, the computer systems are setup to handle these situations in real time (that is people are inbound and will miss their connections, so we need to reassign them fast!), and the equipment swaps and changes made before the flight occurs are an unfortunately by-product.
However in the end, we (the American flying public) made it extremely clear that we consider price the most important element of air travel (reference: Spirit Airline). We are not willing to accept an increase of airfare to pay for a more accurate system, pay the labor costs required for actual people to do the work, or pay for seats that will ensure our party remains together (E+). In essence, we have done this to ourselves.
My recent occurrence of this:
Now to share a story of my recent occurrence of this, Sunday, I was once again checking my Delta (DL) flight to TPA from Portland (PDX) for Thanksgiving, and the cell phone app informed me there was a schedule change. No proactive warning from DL, simply a note by my flight (should I chose to look). Well, they altered the PDX departure time by 10 minutes, cut the connection by 15 minutes (arriving into TPA one minute earlier), changed a flight number on the TPA to ATL flight (causing the seats to be shuffled), and (likely) an equipment swap (change of planes) from ATL to TPA, as we were all reassigned to the row in front of our reservation (same plane type). As such, I spent some time online to get us all settled and check everything. I will continue to be paranoid with all of my family travel, as I understand the game.