  We report measurements of the upper critical field $H_{\mathrm{c2}}$ as
functions of temperature $T$, polar angle $\theta$ (of the field direction with
respect to the crystallographic $c$ axis), and azimuthal angle $\phi$ (of the
field direction relative to the $a$ axis within the $ab$ plane) for the
Cr-based superconductor K$_{2}$Cr$_{3}$As$_{3}$ with a quasi-one-dimensional
and non-centrosymmetric crystal structure. We confirm that the anisotropy in
$H_{\mathrm{c2}}(T)$ becomes inverse with decreasing temperature. At low
temperatures, $H_{\mathrm{c2}}(\theta)$ data are featured by two maxima at
$\theta$ = 0 ($\mathbf{H}\parallel c$) and $\pi/2$ ($\mathbf{H}\bot c$), which
can be quantitatively understood only if uniaxial effective-mass anisotropy and
absence of Pauli paramagnetic effect for $\mathbf{H}\bot c$ are taken
simultaneously into consideration. The in-plane $H_{\mathrm{c2}}(\phi)$ profile
shows a unique threefold modulation especially at low temperatures. Overall,
the characteristic of the $H_{\mathrm{c2}}(\theta, \phi, T)$ data mostly
resemble those of the heavy-fermion superconductor UPt$_3$, and we argue in
favor of a dominant spin-triplet superconductivity with odd parity in
K$_{2}$Cr$_{3}$As$_{3}$.
