When we square the polynomial, we multiply each term by each term. The terms with an odd degree will have an odd degree when multiplied together, so they will still have an odd degree in the final polynomial. The sum of the coefficients of these terms will be the same as the sum of the coefficients of the terms with an odd degree in the original polynomial.

The original polynomial is (2x^3 + 5x^2 - 3x + 1)^2. When we square this polynomial, we get:

(2x^3 + 5x^2 - 3x + 1)^2 = (2x^3 + 5x^2 - 3x + 1)(2x^3 + 5x^2 - 3x + 1)

Multiplying each term by each term, we get:

(2x^3)(2x^3) + (2x^3)(5x^2) + (2x^3)(-3x) + (2x^3)(1) + (5x^2)(2x^3) + (5x^2)(5x^2) + (5x^2)(-3x) + (5x^2)(1) + (-3x)(2x^3) + (-3x)(5x^2) + (-3x)(-3x) + (-3x)(1) + (1)(2x^3) + (1)(5x^2) + (1)(-3x) + (1)(1)

Simplifying each term, we get:

4x^6 + 10x^5 - 6x^4 + 2x^3 + 10x^5 + 25x^4 - 15x^3 + 5x^2 - 6x^4 - 15x^3 + 9x^2 - 3x + 2x^3 + 5x^2 - 3x + 1

Combining like terms, we get:

4x^6 + 20x^5 - 12x^4 - 8x^3 + 14x^2 - 4x + 1

The sum of the coefficients of the terms with an odd degree is 20 + (-8) + (-4) + 1 = $\boxed{9}$.