Is there an example of a category with pullbacks but not equalizers (i.e. at least one pair of parallel morphisms does not have an equalizer)?
Such a category cannot have have the terminal object, it cannot have binary products, there must exist morphisms $f, g$ such that there is no upper bound $u$ such that $u \circ f = u \circ g$, as shown here.
If possible, the example should not be constructed for this purpose.
Let $G$ be a non-trivial group. There is a standard way to regard $G$ as a category with one object, where $G$ is the set of morphisms, and composition is given by the group operation.
Let $g,h\in G$ with $g\neq h$. Then $g$ and $h$ have no equalizer, since there is no element $x\in G$ with $gx=hx$.
But the pullback of $(g,h)$ does exist, given by the pair of elements/morphisms $(g^{-1},h^{-1})$.