All prime numbers are either even or odd, Is it a true statement?

$r:$All prime numbers are either even or odd, Is it a true statement?

I was studying Mathematical Logic then i came across above question. Since here connecting word is "OR" so if i separate two statement then it become

$p:$ All the prime number are even

$q:$ All the prime number are odd

Because both statement $p$ and $q$ are false so final statement $r$ must be false using truth value of statement for "OR" connective.

But my intuition says $r$ is true.

Am i thinking correct?

Please Help me in this.

You erroneously distributed the “all”. The correct interpretation is:

For every prime $p$: $p$ is even or $p$ is odd

Since every integer is either even or odd, we have:

($p$ is even or $p$ is odd) is true for all primes $p$

Therefore it is true — as your intuition suggested

$✗\quad$ Either all numbers are even, or all numbers are odd.

$✔\quad$ All numbers are either even or odd.

In general, $$∀x A(x)\;\text{ or }\;∀x B(x)\quad\text{does not imply}\quad∀x \;\Big(A(x)\text{ or }B(x)\Big);$$ however, $$∃x A(x)\;\text{ or }\;∃x B(x)\quad\text{is equivalent to}\quad∃x \;\Big(A(x)\text{ or }B(x)\Big),$$ and $$∀x A(x)\;\text{ and }\;∀x B(x)\quad\text{is equivalent to}\quad∀x \;\Big(A(x)\text{ and }B(x)\Big).$$