Let $f = -X^4 + X^3Y + Y$ and $g = -X^3 + X^2Y+Y$ and let $C = V(f)$ and $D = V(g)$. Find the points at infinity of the projective completion of $C$ and $D$, $\overline{C}$ and $\overline{D}$, and the asymptotes and parabolic branches of $C$ and $D$
For the first part, the projective completion of $C$ and $D$ are given by $V(F)$ and $V(G)$ where $$F = -X_1^4 + X_1^3X_2 + X_0^3X_2$$ and $$G = -X_1^3 + X_1^2X_2 + X_0^2X_2$$ respectively. So making $X_0 = 0$ in both expressions, I have found the points $(0:0:1)$ with multiplicity $3$ in $\overline{C}$ and $2$ in $\overline{D}$, and $(0:1:1)$, with multiplicity $1$ for both curves.
The problem is that I don't know how to find the asymptotes and parabolic branches of $C$ and $D$. How do I find those? Thank you!
As wikipedia says, asymptotes are often considered only for real curves, and in this answer I will treat $C$ and $D$ as such. Your curves are of a particularly simple type, which makes finding the asymptotes easier.
$C$ is the graph of the rational function $Y=\frac{X^4}{X^3+1}=X-\frac{X}{X^3+1}=X-\frac{X}{(X+1)(X^2-X+1)}$ so $Y=X$ is an oblique asymptote, also $X+1=0$ is a vertical asymptote.
$D$ is the graph of the rational function $Y=\frac{X^3}{X^2+1}=X-\frac{X}{X^2+1}$ so $Y=X$ is an oblique asymptote.
Parabolic branches for graphs of rational functions are when the long division yields a higher degree polynomial than degree 1. Then the graph of the quotient polynomial, like $X$ above, is what the graph of the rational function tends to. If you want to find parabolic branches for more general algebraic curves, the link at wikipedia contains some information.