To find the area of the region enclosed between the parabolas y^2 = 2x - 1 and y^2 = 4x - 3, we need to first determine the points of intersection between the two parabolas.

To find the points of intersection, we can equate the right-hand sides of the two equations:

2x - 1 = 4x - 3

Rearranging the equation, we get:

2x - 4x = -3 + 1

-2x = -2

Dividing both sides by -2, we get:

x = 1

Substituting this value of x back into either of the equations, we can find the corresponding y-coordinate:

y^2 = 2(1) - 1

y^2 = 1

Taking the square root of both sides, we get:

y = ±1

So, the points of intersection between the parabolas are (1, 1) and (1, -1).

To find the area of the region enclosed between the parabolas, we integrate the difference between the two curves with respect to x over the interval where the curves intersect.

First, let's find the equation of the upper curve and the lower curve. From the equations of the parabolas, we can see that y^2 = 4x - 3 represents the upper curve and y^2 = 2x - 1 represents the lower curve.

Since the points of intersection are symmetric about the x-axis, we can integrate from x = 0 to x = 1 and multiply the result by 2 to get the total area.

Using the formula for the area between two curves, the area of the region enclosed between the parabolas is given by:

A = 2∫[0,1] (upper curve - lower curve) dx

= 2∫[0,1] ((4x - 3) - (2x - 1)) dx

= 2∫[0,1] (2x - 2) dx

= 2(∫[0,1] 2x dx - ∫[0,1] 2 dx)

= 2[x^2]0^1 - 2[2x]0^1

= 2(1^2 - 0^2) - 2(2(1) - 2(0))

= 2(1) - 2(2)

= 2 - 4

= -2

Since the area cannot be negative, we take the absolute value:

|A| = |-2| = 2

Therefore, the area of the region enclosed between the parabolas y^2 = 2x - 1 and y^2 = 4x - 3 is 2.

The correct answer is option A.