Remind me never to contribute to an internet discussion forum on any subject save the most uncontroversial or simple (say, gardening). "No, it is not a paradox. Contradictions do not exist." There are two mistakes here.
(1) Mistake one is that the existence of a paradox implies the existence of a contradiction. Not so. A paradox involves an apparent contradiction, i.e. two statements that appear to contradict. Thus a paradox does not imply a contradiction any more than any statement of the form 'it appears that p' implies a statement of the form 'p'. For the antecedent can be true with the consequent false. It can appear that the sun is going round the earth, without the sun going round the earth.
(2) Mistake two is that contradictions do not exist. Not so. A contradiction is two statements, one of which denies the other. Since such pairs of statements exist, it follows that contradictions exist. For example, the statements 'the earth is flat' and 'the earth is not flat' are a contradiction. The statements exist, ergo etc. Probably the claimant meant that no state of affairs expressed by a contradiction exists, which is true, if you believe the Principle of Contradiction.