Given vectors:\[\vec{a} = \hat{i} + \hat{j} + \hat{k}, \quad \vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}, \quad \vec{c} = 7\hat{i} + 9\hat{j} + 11\hat{k}.\]Step 1: Compute the diagonal vectors. The diagonals of the parallelogram are obtained by summing adjacent vectors:\[\vec{d}_1 = \vec{a} + \vec{b} = (1+1)\hat{i} + (1+3)\hat{j} + (1+5)\hat{k} = 2\hat{i} + 4\hat{j} + 6\hat{k},\]\[\vec{d}_2 = \vec{b} + \vec{c} = (1+7)\hat{i} + (3+9)\hat{j} + (5+11)\hat{k} = 8\hat{i} + 12\hat{j} + 16\hat{k}.\]Step 2: Calculate the cross product \( \vec{d}_1 \times \vec{d}_2 \). Using the determinant of a 3x3 matrix:\[\vec{d}_1 \times \vec{d}_2 = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k}
2 & 4 & 6
8 & 12 & 16\end{vmatrix}.\]Expanding the determinant yields:\[\vec{d}_1 \times \vec{d}_2 = \hat{i} \left( 4 \times 16 - 6 \times 12 \right) - \hat{j} \left( 2 \times 16 - 6 \times 8 \right) + \hat{k} \left( 2 \times 12 - 4 \times 8 \right).\]Simplifying the expression results in:\[= \hat{i} (64 - 72) - \hat{j} (32 - 48) + \hat{k} (24 - 32)\]\[= -8\hat{i} + 16\hat{j} - 8\hat{k}.\]Step 3: Calculate the magnitude of the cross product. The magnitude is determined as follows:\[|\vec{d}_1 \times \vec{d}_2| = \sqrt{(-8)^2 + 16^2 + (-8)^2} = \sqrt{64 + 256 + 64} = \sqrt{384} = 8\sqrt{6}.\]Step 4: Calculate the area of the parallelogram. The area is half the magnitude of the cross product:\[\text{Area} = \frac{1}{2} |\vec{d}_1 \times \vec{d}_2| = 4\sqrt{6}.\] Final Answer:\[\boxed{4\sqrt{6} \, \text{sq units}}\]