The equation of a parabola with a vertical axis of symmetry (ie a parabola that opens up or down) is y = ax^2 + bx + c.
But lets go back to the original equation,
x = 3y^2 + 5y - 6
Notice that instead of having y = ax^2 + bx + c, we have a parabola in the form of x = ay^2 + ay + c. This is a parabola with a horizontal axis of symmetry (ie a parabola that opens left or right).
It may help to make a table of x and y values and plug in random numbers to make a rough sketch of the graph.