The "brilliant" commentary from the page is: "Imagine how confused this kid is now when teachers are telling him "Yes, you can get 10 when you add 8+5." It's almost like we are intentionally and structurally trying to make kids less intelligent, or something ..."
This person apparently doesn't even know how much they are demonstrating their lack of understanding about the mathematics they are complaining about. Indeed, they fail to even note that the point they think the teacher is making ("Yes, you can get 10 when you add 8+5") is not what the teacher is actually saying. Indeed, if the parent actually read the entirety of the explanation written in blue marker, they would have seen, "Yes you can. Take 2 from 5 and add it to 8 (8+2=1). Then add 3." (Emphasis mine.) The whole bloody point is that you can get 10 from 8+5, so long as you also understand that you will have an additional 3, because:
8+5 = 10+3
I will admit that it's a really poorly worded question. However, the official answer is actually correct, and - more importantly - it shows the application of the distributive property in arithmetic, which is a principle so foundational in mathematics that - without it - mathematics wouldn't function.
To describe why 8+5 = 10+3 you have to recognize that:
8 = 1+1+1+1+1+1+1+1+1 and 5 = 1+1+1+1+1
8+5 = 1+1+1+1+1+1+1+1+1+1+1+1+1
And this can be re-grouped in any way you want (and those ways go on into infinity if you consider configurations that include negative and non-integer numbers), because of the distributive property. One of those infinite number of ways is 10+3.
A more "algebra" way of thinking about this is to write:
8+5 = x+3
And this sort of thinking is a really important skill when you start to do anything that uses any sort of algebraic thinking (which - in modern terms - means doing almost any sort of function in a spreadsheet).
But let's look at the larger question about the validity of the mathematics in the Common Core. A lot of the Common Core math curriculum is written by mathematicians. Therefore, to challenge the mathematics of the Common Core is almost always going to be baseless and only show that you don't know much about mathematics. In other words, as long the problem isn't about a problem or explanation getting printed incorrectly, the mathematics in the Common Core are almost surely going to be correct.
However, being correct is not the same thing as being worded clearly. Indeed, to challenge the clarity of the teaching materials of the Common Core as being poorly presented or poorly worded can be spot-on, since it is rarely the case that the people who learn how to teach mathematics are actually professional mathematicians (and vice-versa). Therefore, it isn't surprising that the mathematical concepts are not always presented in the manner that makes sense to the non-mathematicians who are teaching the kids (let alone the majority of parents, who haven't sat down with mathematics for decades).
So yeah, the statement "Tell how to make 10 when adding 8+5" may not have been the best way to get people who learned the "old math" (i.e., rote learning that forced people to do pigeon-hole math) to understand or help their children how to actually grasp the powerful, fundamental number theory underlying this simple problem.