In this book here on page 62 parameterised $n$-manifolds are introduced.
The example given is that of a regular curve $\gamma (t) = (X(t),Y(t))$ and a parametrisation $\phi (x,y) := (x, y + Y(x))$. But I think there is a mistake in the book. Say $\phi$ is a local diffeomorphism at $(x_0,0)$ and $U$ is an open set containing $(x_0,0)$. They write:
...it is more convenient if $U$ can always be chosen so that the image $\phi (U)$ contains just one piece of $\gamma$. When this happens we say that the curve is a $1$-manifold.
This should say $2$-manifold, right?
Of course, $\phi$ is the parametrisation and it is a map defined on $U \subset \mathbb R^2$. So we are talking about a $2$-dimensional manifold here.
Edit
(I will upload a screenshot of the relevant page)
No, it's not a typo. The book is defining a "1-manifold" to be a plane curve such that every point in the curve has a neighborhood in $\mathbb{R}^2$ diffeomorphic to $\mathbb{R}^2$, with the curve taken to the $x$-axis. That is, locally, the curve looks like a subspace of $\mathbb{R}^2$. You're essentially throwing away the curves whose closures intersect themselves.
In broader context, the authors are defining a "1-manifold" to be a submanifold of $\mathbb{R}^2$. A subspace $N^n$ of a manifold $M^m$ is a submanifold provided the restrictions to $N$ of coordinate charts on $M$ are coordinate charts on $N$. This is equivalent to saying every point in $N$ may be enclosed by a coordinate neighborhood $U$ in $M$ whose coordinate chart takes $U$ to $\mathbb{R}^m$ and $N\cap U$ to $\{(x_1,x_2,\ldots,x_n,0,0,\ldots,0)\ |\ x_i\in\mathbb{R}\}$.