Using a grammar guide as reference, なくてもいい means "do not have to do ~" or "it is all right if ~".
This is an expression that follows from putting the parts together. The な comes from a negative ない conjugation, which is put into the connective form なくて. も is the usual non-contrastive topic marker: は is like "for this topic, on the other hand", where も is "also for this topic / in this case". いい here is the usual i-adjective "good". "you don't have to do X" is a meaning that comes from smoothing over "it is also good in the case where you don't do X" into something that is actually reasonable to say in English. (Since English doesn't have an intrinsic topic structure, the awkward translation uses extra words to approximate those semantics.)
The ても in 心配しなくても indeed functions like でも which is usually analyzed as a single particle. The trick here is that - at least as far as I've been able to figure out - the で in でも derives from the continuative form of だ, the copula. So, in the same way that XでもYです can mean "it's Y even if it's X", we could use some other verb in place of a copula: XしてもYです "it's Y even after doing X". And we can negate that, per the usual conjugation rules: XしなくてもYです "it's Y even after not doing X", or more naturally, "it's Y even if (you) don't X".
As noted, 心配 has other glosses besides "worry", such as "care for" or "look after". Thus: even if you don't look after the kids (perhaps, help them out with the work?), the work won't be dangerous for them (more literally, "(it) won't be something like dangerous work (for them), so...").
If it instead said 心配しなくてもいい, the いい would complete a predicate, and there would be nothing that works to connect it to the rest. We could possibly parse it as two separate sentences, with the から at the end of the second implying that it's an explanation for the first. "You don't need to worry. After all, it's not dangerous work for them." However, completely dropping a word like that, without some indication of a pause, seems unlikely to me. も followed by more words pretty strongly implies that も is a topic marker for the rest, not for some implied predicate.