When dealing with a block cipher of large block size, the text appearing as a quote in the question
Most modern block ciphers have a 128-bit block size, but they operate on 32-bit words. They build the encryption function from many 32-bit operations.This has proved to be a very successful method, but it has one side effect. It is rather hard to build an odd permutation from small operations; as a result,virtually all block ciphers only generate even permutations.
is about a merely theoretical attack. The book's chapter 3 acknowledges it:
This attack has no practical significance whatsoever.
That's because knowing the parity of the permutation implemented by an otherwise ideal block cipher with a $b$-bit block and some fixed key only helps adversaries after they have obtained $2^b-2$ plaintext/ciphertext pairs: the last two plaintext/ciphertext pairs are revealed by that parity. Before that threshold, nothing actionable comes from that one bit of information.
For example, with $b=3$, after an adversary obtained plaintext/ciphertext pairs 0/1, 1/6, 2/5, 3/0, 4/2, 5/7, which we can picture as
0 1 2 3 4 5 6 7
1 6 5 0 2 7 ? ?
and if the permutation is known to be even, then the adversary can determine¹ that the remaining pairs are 6/3, 7/4 (rather than 6/4, 7/3). But before the adversary obtained the pair 5/7, knowing that the permutation was even was of no help to predict if 5 maps to 3, 4, or 7.
Even if a 128-bit block cipher is known to implement an even permutation for any key, that's not an exploitable weakness. It does allow to build a theoretical distinguisher from an ideal cipher, but only after making so many queries ($2^{128}-1$) to the encryption or decryption oracle² that it does not count as attack against usual or reasonable security definitions.
The quoted text is somewhat exaggerating the difficulty of building an odd permutation from small operations; see these comments by poncho:
The standard trick for small block Feistel ciphers is to use modular addition, rather than xor, in each round; that way, the round, and hence the permutation, has a 0.5 probability of being odd. (…) If the two halves of the Feistel state are $a, b$, then the update $a\gets a+F(k,b)$ can be odd; in fact, it will be if an odd number of the $F(k,b)$ values are odd (fixed $k$, over all possible values of $b$).
¹ Proof: going from 01234567 to 16502734 can be done with an even number of permutations, e.g. 01234567 → 10234567 → 16234507 → 16534207 → 16504237 → 16502437 → 16502734.
² The challenger randomly choose an ideal random cipher or an even random cipher, the distinguisher tries to guess that choice. It needs $2^{128}-1$ queries to determine if the cipher is even or odd, if odd outputs 'ideal', otherwise outputs 'even'. It succeeds with probability $3/4$.
