To determine the correct increasing order of bond length among the given species ($O_2^{+}$, $O_2$, $O_2^{-}$, and $O_2^{2-}$), we can use Molecular Orbital (MO) theory to calculate the bond order for each species.

The relationship between bond order and bond length is inversely proportional:
$\text{Bond Length}\propto \frac{1}{\text{Bond Order}}$
This means that a higher bond order corresponds to a shorter bond length, and a lower bond order corresponds to a longer bond length.

Let us first write down the molecular orbital configuration for the oxygen molecule ($O_2$), which has 16 electrons:
$\sigma 1s^2{\sigma }^{*}1s^2\sigma 2s^2{\sigma }^{*}2s^2\sigma 2p_z^2(\pi 2p_x^2=\pi 2p_y^2)({\pi }^{*}2p_x^1={\pi }^{*}2p_y^1)$

The formula for calculating Bond Order (BO) is:
$\text{BO}=\frac{N_b-N_a}{2}$
where $N_b$ is the number of bonding electrons and $N_a$ is the number of antibonding electrons.

Now, we calculate the bond order for each given species:

1. For $O_2^{+}$ (15 electrons):
One electron is removed from the antibonding molecular orbital (${\pi }^{*}2p_x$ or ${\pi }^{*}2p_y$).
Number of bonding electrons, $N_b=10$
Number of antibonding electrons, $N_a=5$
$\text{Bond Order of}{O}_2^{+}=\frac{10-5}{2}=2.5$

2. For $O_2$ (16 electrons):
Number of bonding electrons, $N_b=10$
Number of antibonding electrons, $N_a=6$
$\text{Bond Order of}{O}_2=\frac{10-6}{2}=2.0$

3. For $O_2^{-}$ (17 electrons):
One electron is added to the antibonding molecular orbital.
Number of bonding electrons, $N_b=10$
Number of antibonding electrons, $N_a=7$
$\text{Bond Order of}{O}_2^{-}=\frac{10-7}{2}=1.5$

4. For $O_2^{2-}$ (18 electrons):
Two electrons are added to the antibonding molecular orbitals.
Number of bonding electrons, $N_b=10$
Number of antibonding electrons, $N_a=8$
$\text{Bond Order of}{O}_2^{2-}=\frac{10-8}{2}=1.0$

Comparing the calculated bond orders:
$O_2^{+}\text{(2.5)}>O_2\text{(2.0)}>O_2^{-}\text{(1.5)}>O_2^{2-}\text{(1.0)}$

Since bond length is inversely proportional to bond order, the increasing order of bond length is the reverse of the bond order sequence:
$O_2^{+}<O_2<O_2^{-}<O_2^{2-}$

Therefore, the correct increasing order of bond length is:
$O_2^{+},O_2,O_2^{-},O_2^{2-}$