Let us see one by one :
a) L1 is parallel to L1 i.e. itself hence it is a reflexive relation
b) If L1 is parallel to L2 , then L2 is also parallel to L1 , so it is a symmetric relation..
c) If L1 is parallel to L2 , L2 is parallel to L3 , then L1 is also parallel L3 as per the geometry , so the relation is also transitive..
d) For antisymmetric to hold we can think it as in xRy , x = y is allowed but if x != y then yRx is not allowed..But here if L1 R L2 holds then L2 R L1 also holds for set of lines and L1 and L2 are distinct..Hence it is not antisymmetric..
e) For asymmetric to hold not even same ones allowed that is , xRx is also not allowed besides antisymmetric condition..But here L1 R L1 is allowed and we have seen it is not anti symmetric also..Hence it is not asymmetric relation..