Linear maps preserving parallel matrix pairs with respect to the Ky-Fan $k$-norm

Two bounded linear operators $A$ and $B$ are parallel with respect to a norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ if $\mathrm{\parallel }A+\mu B\mathrm{\parallel }=\mathrm{\parallel }A\mathrm{\parallel }+\mathrm{\parallel }B\mathrm{\parallel }$ for some scalar $\mu$ with $\mathrm{\mid }\mu \mathrm{\mid }=1$. Characterization is obtained for bijective linear maps sending parallel bounded linear operators to parallel bounded linear operators with respect to the Ky-Fan $k$-norms.