Since both diagonals are perpendicular, it must be a rhombus. It also may be a square, but equal sides are not given.

To find the equations of the diagonals of a parallelogram, we need to find the coordinates of its vertices.

Let's assume that the coordinates of P are (a, b).

Since the diagonals of a parallelogram bisect each other, the midpoint of the diagonal PR is also the midpoint of QS. Therefore, the coordinates of the midpoint of PR are [(a+6)/2, (b-3)/2], and this point lies on the line 3y = 4x.

So, we have:
3(b-3)/2 = 4(a+6)/2
3b - 9 = 4a + 24
3b = 4a + 33
b = (4/3)a + 11

Similarly, since the diagonals of a parallelogram bisect each other, the midpoint of the diagonal PS is also the midpoint of QR. Therefore, the coordinates of the midpoint of PS are [(a+6)/2, (b+3)/2], and this point lies on the line 6x + 5y = 20.

So, we have:
6(a+6)/2 + 5((b+3)/2) = 20
3a + 18 + 5b + 15 = 40
3a + 5b = 7

Now, we have a system of equations:
b = (4/3)a + 11
3a + 5b = 7

To solve this system, we can substitute the value of b from the first equation into the second equation:

3a + 5((4/3)a + 11) = 7
3a + (20/3)a + 55 = 7
9a + 20a + 165 = 21
29a = -144
a = -144/29

Substituting this value back into the first equation, we can find b:

b = (4/3)(-144/29) + 11
b = -192/29 + 319/29
b = 127/29

Therefore, the coordinates of P are (-144/29, 127/29).

To find the coordinates of Q, R, and S, we can use the fact that the opposite sides of a parallelogram are parallel. Therefore, the coordinates of Q, R, and S can be found by shifting the coordinates of P by the same amount horizontally and vertically.

The horizontal shift is the difference between the x-coefficients of the two given lines:
6 - 3 = 3

The vertical shift is the difference between the y-coefficients of the two given lines:
0 - (-3) = 3

Therefore, the coordinates of Q are:
Q: (-144/29 + 3, 127/29 + 3) = (-144/29 + 87/29, 127/29 + 87/29) = (-57/29, 214/29)

Similarly, the coordinates of R are:
R: (-144/29 + 3, 127/29) = (-144/29 + 87/29, 127/29) = (-57/29, 127/29)

And the coordinates of S are:
S: (-144/29, 127/29)

Therefore, the coordinates of