  In this paper, we classify the 1-Weierstrass points of the Kuribayashi
quartic curves with two parameters $a$ and $b$ defined by the equation
\[C_{a,b}:x^{4}+y^{4}+z^{4}+ax^{2}y^{2}+b(x^{2}+y^{2})z^{2}=0,\] such that
$(a^2-4)(b^2-4)(a^2-1)(b^{2}-a-2)\neq0.$ Furthermore, the geometry of these
points is investigated.
