considering the values of c as 1369(i.e. 37*37) and d as 361(i.e. 19*19)
=> 1369-361=1008

That is one scenario, but how do you find 37 and 19, except by trying various numbers? I found 33 and 9 fit as well, by making a lucky guess.

SOLUTION #1:
All odd integers can be rewritten as 2k + 1, where k is an integer.
So, if c is the square of an odd integer, we can write: c = (2k + 1)² for some integer k.
Likewise, we can write: d = (2j + 1)² for some integer j.

So, c - d = (2k + 1)² - (2j + 1)²
Expand: c – d = (4k² + 4k + 1) - (4j² – 4j + 1)
Simplify: c – d = 4k² + 4k - 4j² – 4j
Factor to get: c – d = 4(k² + k - j² – j)
This tells us that c – d is a multiple of 4
Since answer choice A is the only multiple of 4, it must be the correct answer.

ASIDE: To determine whether a number is divisible by 4, just examine the number created by the last two digits.
So, we know that 1008 is divisible by 4 because 08 is divisible by 4
Conversely, we know that 1018 is NOT divisible by 4 because 18 is NOT divisible by 4
Likewise, we know that 1022 is NOT divisible by 4 because 22 is NOT divisible by 4
etc.


SOLUTION #2:
If c and d are the squares of odd integers, we can let c = x² and let d =y²(where x and y are odd integers)

Now take: c - d
And rewrite as: x² – y²
Factor: (x + y) (x - y)

Since x and y are both odd, we can rewrite the above expression as follows: (odd + odd) (odd - odd)
Simplify to get: (some EVEN integer)(some other EVEN integer)

If an integer is EVEN, we can rewrite it as 2k, where k is some integer.
So, we can take (some EVEN integer)(some other EVEN integer) and rewrite it as: (2j)(2k), where j and k are integers.
Finally we can rewrite (2j)(2k) as 4jk

We’ve now taken c - d and rewritten it as 4jk, where j and k are integers.
Since 4jk is a multiple of 4, it must be the case that c - b is a multiple of 4.

When we check the answer choices, we see that only answer choice A is a multiple of 4.
Answer: A


SOLUTION #3:
Even if you didn't identify an algebraic solution, you can still test some possible values and look for a pattern.
Since c and d are the squares of odd integers, it could be the case that c = 9² = 81, and d =7² = 49, in which case c - d = 81 - 49 = 32
Or it could be the case that c = 9² = 81, and d =5² = 25, in which case c - d = 81 - 25 = 56
Or it could be the case that c = 9² = 81, and d =3² = 9, in which case c - d = 81 - 9 = 72
Or it could be the case that c = 9² = 81, and d =1² = 1, in which case c - d = 81 - 1 = 80
Or it could be the case that c = 11² = 121, and d =9² = 81, in which case c - d = 121 - 81 = 40
At this point, we might recognize that 32, 56, 72, 80 and 40 are all divisible by 4, in which case it could be true that the correct answer is also divisible by 4.
Answer: A

Aside: As you might guess, SOLUTION #3 isn't the most precise way to approach this question, but it still gives you a reasonable chance of identifying the correct answer

(2k+1)^2 is not 4k^2 -4k + 1, it is 4k2+4k+1

Good catch!
I've edited my response.

KUDOS for you!