Step-by-step Solution:

To determine which points lie on the parabola, we need to check if their coordinates satisfy the given equation. A point \((x, y)\) lies on the curve if substituting its \(x\) and \(y\) values into the equation results in a true statement.

The equation of the parabola is given as: \(y^2 = 4x\).

We will test the points from each option.

1. Test Option A: (4, 4) and (4, -4)
For the point (4, 4):
Substitute \(x=4\) and \(y=4\) into the equation.
Left Hand Side (LHS): \(y^2 = (4)^2 = 16\).
Right Hand Side (RHS): \(4x = 4(4) = 16\).
Since LHS = RHS (\(16=16\)), the point (4, 4) lies on the parabola.

For the point (4, -4):
Substitute \(x=4\) and \(y=-4\) into the equation.
LHS: \(y^2 = (-4)^2 = 16\).
RHS: \(4x = 4(4) = 16\).
Since LHS = RHS (\(16=16\)), the point (4, -4) also lies on the parabola.
Conclusion: Option A is the correct answer as both points satisfy the equation.

2. Test Other Options (for verification):
Option B: Checking point (3, 5)
LHS: \(y^2 = 5^2 = 25\).
RHS: \(4x = 4(3) = 12\).
Since \(25 \neq 12\), this option is incorrect.

Option C: Checking point (1, 3)
LHS: \(y^2 = 3^2 = 9\).
RHS: \(4x = 4(1) = 4\).
Since \(9 \neq 4\), this option is incorrect.

Option D: Checking point (6, 8)
LHS: \(y^2 = 8^2 = 64\).
RHS: \(4x = 4(6) = 24\).
Since \(64 \neq 24\), this option is incorrect.

Result:
The points in option A, (4,4) and (4,-4), are the only pair that both lie on the parabola \(y^2=4x\).