Answer
parabola. vertex $V(1, -\frac{13}{6})$, focus $F(1,-\frac{5}{3})$, directrix $y=-\frac{8}{3}$
See graph.
Work Step by Step
Step 1. Identify the type of the conics: rewrite the equation as $3(x^2-2x+1)=6y+10+3$ or $(x-1)^2=2(y+\frac{13}{6})$, and the equation is of the form of a parabola.
Step 2. The vertex can be found at $V(1, -\frac{13}{6})$
Step 3. With $4p=2$, we get $p=\frac{1}{2}$, so the focus is at $(1, p-\frac{13}{6})$ or $F(1,-\frac{5}{3})$
Step 4. The directrix can be found as $y=-\frac{13}{6}-p=-\frac{8}{3}$
Step 5. See graph.
