Given vectors DA = a and AB = b, with AB parallel to DC and DC = 2AB, express DB, DC, and BC in terms of a and b.

To go from D to B, we can go from D to A, then from A to B.

$$\vec{DB} = \vec{DA} + \vec{AB}$$

$$\vec{DB} = \mathbf{a} + \mathbf{b}$$

The vector $\vec{DB}$ is $\boxed{\mathbf{a} + \mathbf{b}}$.

$$\vec{DC} = 2\mathbf{b}$$

The vector $\vec{DC}$ is $\boxed{2\mathbf{b}}$.

To go from B to C, we can go from B to D, then from D to C.

$$\vec{BC} = \vec{BD} + \vec{DC}$$

Note that $\vec{BD} = -\vec{DB}$.

We found $\vec{DB} = \mathbf{a} + \mathbf{b}$ and $\vec{DC} = 2\mathbf{b}$.

$$\vec{BC} = -(\mathbf{a} + \mathbf{b}) + 2\mathbf{b}$$

$$\vec{BC} = -\mathbf{a} - \mathbf{b} + 2\mathbf{b}$$ $$\vec{BC} = -\mathbf{a} + \mathbf{b}$$

The vector $\vec{BC}$ is $\boxed{-\mathbf{a} + \mathbf{b}}$.