The increasing replacement of conventional generators by variable renewable energy sources is reducing the flexibility of the power system, and consequently reducing its reliability indexes. To compensate for this reduction of flexibility, market participation of aggregators of multi-energy systems has been proposed in the literature. Under this scope, this paper presents a network-secure bidding optimization strategy to assist aggregators of multi-energy systems calculating electricity (energy and reserve), gas and carbon bids, considering multi-energy network constraints. This strategy is a distributed approach based on the alternating direction method of multipliers, where the aggregator collaborates with the operators of electricity, gas and heat networks to calculate network-secure bids. The proposed strategy is benchmarked against two other approaches. The results show that the newly developed strategy computes multi-energy and network-secure bids with execution times that suit the timelines of the electricity, gas, and carbon markets. The joint optimization of multi-energy systems reduced the aggregator's costs by 89% compared to a single energy-vector approach. Furthermore, two sensibility studies were also performed. The first study revealed that in the presence of slow ramp-rate resources (e.g. combined heat and power systems), aggregator's costs can decrease up to 87% when considering slower response times to the secondary reserve signal. In the second study, it was observed that the bidding behavior of the aggregator only starts changing significantly with carbon prices higher than 200euro/tCO2.