1. Use resistance formula

For a wire of length $L$ and radius $r$, resistance is $R=\rho\frac{L}{A}=\rho\frac{L}{\pi r^2}$ where $\rho$ is the resistivity.

So, $R \propto \frac{L}{r^2}$ for wires of the same material.

2. Given ratios

Let the two wires be 1 and 2.

Length ratio: $\frac{L_1}{L_2}=\frac{4}{3}$ Radius ratio: $\frac{r_1}{r_2}=\frac{2}{3}$

3. Find resistance ratio

Since $R \propto \frac{L}{r^2}$ we get $\frac{R_1}{R_2}=\frac{L_1}{L_2}\cdot\frac{r_2^2}{r_1^2}$

Substitute the given ratios: $\frac{R_1}{R_2}=\frac{4}{3}\cdot\frac{(3)^2}{(2)^2}$ $\frac{R_1}{R_2}=\frac{4}{3}\cdot\frac{9}{4}=3$

Thus, $R_1:R_2=3:1$

4. Use parallel connection condition

In parallel, both wires have the same potential difference. Hence current is inversely proportional to resistance: $I \propto \frac{1}{R}$

Therefore, $\frac{I_1}{I_2}=\frac{R_2}{R_1}=\frac{1}{3}$

So the ratio of currents is $I_1:I_2=1:3$

5. Match with options

The correct option is: $\boxed{\frac{1}{3}}$ which is Option B.