The formula for the perpendicular distance between parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is:
\[\n\text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}\n\]
For the lines \( 3x + 4y - 6 = 0 \) and \( 6x + 8y + 7 = 0 \), we have \( A = 3 \), \( B = 4 \), \( C_1 = -6 \), and \( C_2 = 7 \).
Calculate the distance using the formula:
\[\n\text{Distance} = \frac{|7 - (-6)|}{\sqrt{3^2 + 4^2}} = \frac{|7 + 6|}{\sqrt{9 + 16}} = \frac{13}{\sqrt{25}} = \frac{13}{5} = \frac{19}{10}\n\]
Therefore, the perpendicular distance between the lines is \( \frac{19}{10} \) units.