Linear Speedup and Amdahl's law
The phrase

The speedup we achieve is linear to the number of cores.

refers only to the C portion of the program. Therefore the running time on $N$ cores is
$$ t_A + t_B + \frac{t_C}{N}. $$
The speedup is
$$
\frac{t_A + t_B + t_C}{t_A + t_B + t_C/N} =
\frac{1}{\frac{t_A + t_B + t_C/N}{t_A+t_B+t_C}} =
\frac{1}{\frac{t_A+t_B}{t_A+t_B+t_C}+\frac{1}{N} \frac{t_C}{t_A+t_B+t_C}}.
$$
This is Amdahl's law.