The compound(s) having peroxide linkage is(are)

Peroxide (peroxo) linkage means the presence of an $$\mathrm{O\!-\!O}$$ single bond inside the molecule.
Therefore, we must write the accepted structures of the given oxo-acids of sulphur and look for an $$\mathrm{O\!-\!O}$$ bond.

Option A : $$H_2S_2O_7$$ (pyrosulphuric acid, oleum)
$$H_2S_2O_7 : \; HO\!-\!S(=O)_2\!-\!O\!-\!S(=O)_2\!-\!OH$$
The two sulphur atoms are bridged by a single oxygen $$\,(S\!-\!O\!-\!S)$$, but there is no $$\mathrm{O\!-\!O}$$ bond.
Hence, it does not contain a peroxide linkage.

Option B : $$H_2S_2O_8$$ (peroxodisulphuric acid, Marshall’s acid)
$$H_2S_2O_8 : \; HO\!-\!S(=O)_2\!-\!O\!-\!\mathbf{O}\!-\!S(=O)_2\!-\!OH$$
A peroxide bridge $$\mathbf{(O\!-\!O)}$$ joins the two $$SO_3$$ groups, so the molecule definitely contains a peroxide linkage.

Option C : $$H_2S_2O_5$$ (disulphurous acid)
An accepted structure is $$HO\!-\!S(=O)\!-\!O\!-\!S(=O)\!-\!OH$$ (or an $$S\!-\!S$$ bonded form in equilibrium).
Neither form possesses an $$\mathrm{O\!-\!O}$$ bond, so no peroxide linkage is present.

Option D : $$H_2SO_5$$ (peroxomonosulphuric acid, Caro’s acid)
$$H_2SO_5 : \; HO\!-\!\mathbf{O}\!-\!S(=O)_2\!-\!OH$$
One hydroxyl oxygen is linked to the sulphur through an $$\mathbf{O\!-\!O}$$ bond; hence a peroxide linkage is present.

Thus, the compounds that contain a peroxide linkage are:

Option B which is: $$H_2S_2O_8$$,  Option D which is: $$H_2SO_5$$