There is a systematic method to solve problems of this sort (the $p,q$ method) by reducing them to computations which can be done by hand. Assume the vertices are $(0,0), (1,0), (p,q)$. The three new points have the form $(x+t(p-1),y+tq), (x-up,y-uq), (x+s,y)$ with three parameters.These can be reduced to one free parameter by the parallelism condition. One then uses the condition on the rank of the matrix with rows $$(x^2 ,xy , y^2 ,x ,y, 1)$$ one for each of the seven points, which ensures that they line on a conic. Now compute the quadratic equation of the corresponding conic (with coefficients which depend on $p,q,x,y$). It is then routine to compute the coordinates of $O$ and verify the collinearity condition.
This is, of course, not as elegant as a (presumable) synthetic proof but it has the advantage of placing the result in a general context. It often suggests possible refinements (in this case I would try replacing the parallelism condition by suitable restraints on the angles between the three directions).
