It seems like your problem is here:

                    if (collision == GameMap.TileCollision.Block || IsOnGround)
                    {
                        // Resolve the collision along the Y axis.
                        Position = new Vector2(Position.X, Position.Y + depth.Y);

                        // Perform further collisions with the new bounds.
                        bounds = BoundingRectangle;
                    }

You should take away depth.Y from Position.Y, as co-ordinates in game systems typically originate from (0,0) being the top left of the map.

In addition, you should check to see which edge you should push the player to depnding on which is less deep, instead of just pushing higher Y values up and higher X values to the right.

If I understand correctly, your movement issue can be resolved by David's answer, but you want to implement being able to move upwards through platforms. This can be solved with an additional if statement. When you collide with a platform, check the players Y velocity and the target block type to see if you can move upwards through it. If the player's Y velocity is set above 0, or the block type cannot be moved through upwards, then resolve the collision as normal.

It seems like your problem is here:

                    if (collision == GameMap.TileCollision.Block || IsOnGround)
                    {
                        // Resolve the collision along the Y axis.
                        Position = new Vector2(Position.X, Position.Y + depth.Y);

                        // Perform further collisions with the new bounds.
                        bounds = BoundingRectangle;
                    }

You should take away depth.Y from Position.Y, as co-ordinates in game systems typically originate from (0,0) being the top left of the map.

It seems like your problem is here:

                    if (collision == GameMap.TileCollision.Block || IsOnGround)
                    {
                        // Resolve the collision along the Y axis.
                        Position = new Vector2(Position.X, Position.Y + depth.Y);

                        // Perform further collisions with the new bounds.
                        bounds = BoundingRectangle;
                    }

You should take away depth.Y from Position.Y, as co-ordinates in game systems typically originate from (0,0) being the top left of the map.

In addition, you should check to see which edge you should push the player to depnding on which is less deep, instead of just pushing higher Y values up and higher X values to the right.