Let's denote the total number of parks as \( P \).
According to the report, \( \frac{1}{3} \) of the parks have flagpoles. Therefore, the number of parks with flagpoles is:
\[ \frac{1}{3} P \]
The city then decides to install flagpoles in another \( \frac{1}{3} \) of the parks. The number of parks in this additional \( \frac{1}{3} \) is:
\[ \frac{1}{3} P \]
To find the total number of parks with flagpoles after the installation of the additional flagpoles, we add the number of parks that already have flagpoles to the number that will receive new flagpoles:
\[ \text{Total parks with flagpoles} = \frac{1}{3} P + \frac{1}{3} P = \frac{2}{3} P \]
To find the total fractional portion of parks that have flagpoles compared to the total number of parks, we divide the number of parks with flagpoles by the total number of parks:
\[ \text{Fraction of parks with flagpoles} = \frac{\frac{2}{3} P}{P} = \frac{2}{3} \]
Therefore, the total fractional portion of parks that have flagpoles is
\[ \boxed{\frac{2}{3}}. \]