We study a class of second-order degenerate linear parabolic equations in divergence form in
(
−
∞
,
T
)
×
R
+
d
(-\infty , T) \times {\mathbb {R}}^d_+
with homogeneous Dirichlet boundary condition on
(
−
∞
,
T
)
×
∂
R
+
d
(-\infty , T) \times \partial {\mathbb {R}}^d_+
, where
R
+
d
=
{
x
∈
R
d
:
x
d
>
0
}
{\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\}
and
T
∈
(
−
∞
,
∞
]
T\in {(-\infty , \infty ]}
is given. The coefficient matrices of the equations are the product of
μ
(
x
d
)
\mu (x_d)
and bounded uniformly elliptic matrices, where
μ
(
x
d
)
\mu (x_d)
behaves like
x
d
α
x_d^\alpha
for some given
α
∈
(
0
,
2
)
\alpha \in (0,2)
, which are degenerate on the boundary
{
x
d
=
0
}
\{x_d=0\}
of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.