  Rizzo showed that the family of elliptic curves $\mathcal{W}(t)
:y^2=x^3+tx^2-(t+3)x+1$, a well-known example of Washington, has root number
$W(\mathcal{W}(t))=-1$ for all $t\in\mathbb{Z}$. In this paper we generalize
this example and identify the families of small degree on which this phenomenon
happens. Motivated by results from David, Bettin and Delaunay
(arXiv:1612.03095) and Desjardins (arXiv:1810.12787), we study in detail the
two families $\mathcal{F}_s(t):y^2=x^3+3tx^2+3sx+st$ and
$\mathcal{L}_{w,s,v}(t): wy^2=x^3+3(t^2+v)x^2+3sx+s(t^2+v)$ and describe
necessary and sufficient conditions for which subfamilies of $\mathcal{F}_s(t)$
have constant root number on integer fibres. We further prove similar but
partial results on $\mathcal{L}_{w,s,v}(t)$. Our results give examples of
subfamilies for which there is rank elevation at integer fibres.
