Given the diagram here, we can easily trace out paths of light rays. But if I imagine a stationary particle moving upwards in time going through the center of the light cone, it appears as if it doesn't move in space. Whereas I would expect it to be drawn towards the black hole by gravity. (In a parabolic curve, at least far away from the horizon).
How do we estimate the movement of a massive particle not on the light cone using this diagram?
The diagram shows the spacetime in Schwarzschild coordinates $(t, r)$ which describe what measured by an observer at infinity (far away from the horizon). Schwarzschild coordinates fail to show what happens in the interior region of a black hole; in fact a photon or a massive particle in free fall are observed to progressively slow down as they approach the horizon until they eventually freeze on it.
To understand what happens, you have to change coordinates and pass to Eddington–Finkelstein coordinates or to Kruskal–Szekeres coordinates. In reality a photon or a massive particle in free fall would go through the horizon until they hit the physical singularity at the center of the black hole.
Coming back to the diagram in Schwarzschild coordinates, a massive particle follows a timelike path, so it is inside the light cone. As the light cone closes up progressively when approaching the horizon, neither the photon nor the massive particle will go through the horizon. This is the describing limit of Schwarzschild coordinates.
Note: I think your mention of a stationary particle is confusing. If a particle is stationary, by definition it remains at a constant radial coordinate. However that is not a geodesic, but an accelerated path. If you have a spacecraft to remain stationary in the spacetime geometry around the horizon, it has to use its engines to offset the curvature.
