What's the POD?
One thing needed is better preparation on the part of Paraguay (training/weaponry/etc) and a better navy so that they can take control of the river, the banks, and the mouth. Controlling the river is absolutely paramount. After that, it's a matter of knocking the triple alliance out. Uruguay is pretty much a non consideration. If Argentina and Brazil drop out, U is sunk. Brazil is a giant, but the center is far away, so as long as Paraguay doesn't seek too much Brazilian land, Brazil may get tired and drop out. Most of what Paraguay is going to gain is at the expense of Argentina and Uruguay (probably entre rios, corrientes, missiones, maybe Uruguay, and possibility of Brazil's rio do sul). That makes a big country on Buenos Aires doorstep, and Argentina is not going to drop out easily if there's that kind of threat so close. Maybe if P just kicks some butt and then settles for minor gains, the TA would accept it, but why bother starting a war which is intended to establish yourself as a regional power and then not get enough out of it to make yourself a power.
So, you have Paraguay with a beach head at the mouth of the river, thus access to the outside world, and an Argentina not likely to just roll over and die, and a Brazil that isn't likely to drop out too fast either. Paraguay needs a massive knockout punch to all of the combatants, and that's a bit unrealistic. However, with a real regional war going on, instead of the OTL long siege, European powers are going to sit up and take notice and likely look to mediate.