For a system of linear equations to have no solution, the lines they represent must be parallel but distinct. The condition for two linear equations, $Ax + By = C$ and $Dx + Ey = F$, to be parallel is that the ratio of their x-coefficients equals the ratio of their y-coefficients. In this case, we have:
\[ \frac{p}{-4} = \frac{3}{k} \]
This simplifies to:
\[ p \cdot k = -12 \quad (1) \]
For the lines to not coincide (i.e., to be distinct), the ratio of the constant terms must be different from the ratio of the coefficients. Therefore, we must have:
\[ \frac{2}{p} \ne \frac{a}{3} \]
This inequality leads to:
\[ 2a + k \ne 0 \quad (2) \]
Hence, the system has no solution if $2a + k \ne 0$, which matches Option (4).