The two plates act as a particle filter in the middle (virtual photons with longer wavelengths can't form), creating a pressure differential as the photons and whatnot impact the plates from both sides. Because there are fewer virtual particles forming between the plates due to the filtering effect, the average force applied is smaller.
What a lot of people don't know is that conservation of mass/energy is something of a white lie. Conservation of mass/energy is conserved over time but is not required to be conserved at any given instant in time. For various reasons having to do with the caveats that go along with it, we don't see this manifest meaningfully in ordinary day-to-day life. There is nothing in principle that prevents a Ferrari from materializing in my driveway out of nothing, but I'm not counting on it. The odds are probably statistically better that my truck spontaneously rearranges itself into a Ferrari due to a highly improbable thermodynamic event.
Think of the universe as a bank. You can "borrow" things into existence from nothing, but the more you borrow, the stiffer the terms, and on average, the sooner you have to pay it back.
Now listen, you who say, Today or tomorrow we will go to this or that city, spend a year there, carry on business and make money. Why, you do not even know what will happen tomorrow. What is your life? It is a mist that appears for a little while and then vanishes. Instead, you ought to say, If it is the Lord's will, we will live and do this or that. As it is, you boast and brag. All such boasting is evil. Anyone, then who knows the good he ought to do and doesn't do it, sins. James 4:13-17
I take it you didn't have a source to support your something from nothing addendum to the Casimir Effect?