The method consists of a polymerization process or a copolymerization process which is carried out in two parallel reactors, and each of dienes are polymerized or copolymerized with each other and/or with arylvinyl compounds, wherein an organolithium initiator, an electron donor additive and a branching agent are fed into the first reactor, and an organo lithium initiator, an electron donor additive and a functionalizing agent are fed into the second reactor, and the resultant polymerized mixtures are subsequently mixed with each other. This method allows for producing branched functionalized (co)polymers of dienes that are characterized by a statistical distribution of monomer units, a high content of vinyl units (1,2-butadiene units and/or 3,4-isoprene units (more than 60%)), a narrow molecular weight distribution and a controllable content of branched and functionalized parts in the (co)polymer.