In the book of Analysis on Manifolds by Munkres, at page 189, it is given that
(For a parametrised-manifold $Y_\alpha$)
However, this definition suggests that there are integral with are not taken wrt to the volume.
My question is why do we need to specify that we are taking integral wrt volume in the parametrised-manifold case, which was not something that we did not needed when we defined integral over the region in $\mathbb{R}^k$ ?
Secondly, what are the other integral that are not taken wrt to the volume element ?
That is the natural definition of an integral over a parameterized manifold and the notation that Munkres happens to like.
This is just the generalization of the familiar notion of surface integral in three-dimensional calculus.
I'm sure you recall for a surface $Y_\alpha = \{\mathbf{\alpha}(s,t): (s,t) \in \,$A$\, \subset \mathbb{R}^2\} \subset \mathbb{R}^3$ where $\mathbf{\alpha} \in C^1(A)$, the surface integral of a function $f:Y_\alpha \to \mathbb{R}$ is defined as
$$\int_{Y_\alpha} f \, dS = \iint_A f(\mathbf{\alpha}(s,t)) \left\|\frac{\partial \mathbf{\alpha}}{\partial s} \times \frac{\partial \mathbf{\alpha}}{\partial t}\right\|\, ds \, dt$$
Just compare the corresponding terms to what appears in Munkres' equation and note that the RHS is something altogether different than $\displaystyle\iint_A f(\mathbf{\alpha}(s,t)) \, ds \, dt.$ Also the term "volume" is used in the general sense for sets in $\mathbb{R}^n$. You can substitute the word "area" when appropriate.
The integral over the parameterized manifold has to be defined just as the area of a rectifiable set $A \subset \mathbb{R}^n$ is defined by the Riemann integral $\displaystyle \int_Q \chi_A$, where $Q$ is an enclosing "rectangle" in $\mathbb{R}^n$. Such a definition makes sense in that it recovers what we all expect as the area of a simple rectangle: $\int_{[0,l] \times [0,w]} f = l \, w$ when $f = 1$.