There are a number of issues here which you should be aware of, and which may help you better work with your wife when raising your child:
First, children understand from a very early age that each parent does things differently, and will go to the parent that will give them what they want or need, and will even behave very differently based on the parent they are with. This isn't a problem, necessarily, however it does mean that you two are individuals, and will each have a different relationship with your child than the other. This may be beneficial in the long run, as long as you are generally united on the things you value and believe in. Having the same values and beliefs come from two people with two perspectives can cement those values and beliefs more strongly than otherwise.
However, your child will use these differences to manipulate situations. As long as this isn't egregious I wouldn't worry about it, just insofar as you are making sure no abuse is occurring and it isn't unintentionally teaching them selfish patterns that could become stumbling blocks in the future. It's common for humans to seek out the easiest path to a goal, whether in work, school, or at home, so this isn't intrinsically a bad thing, as long as it isn't extreme.
Second, discipline is often one aspect of respect. As long as your wife has some boundaries that are apparent, or if she gains the child's respect for her thoughts and expectations in other ways, then this won't be an issue.
However I've seen all too many families where your dynamic is in play, and the children walk all over the non-disciplining parent. This may not seem problematic if the parent is willing to plead and beg daily for their children to meet expectations, but the real problem occurs later in life when the youth ignores that parent's counsel in important matters, and only listens to the other parent whom they've learned to respect. The non-disciplining parent finds themselves feeling left out, or of little consequence.
Again, discipline is just one aspect of a strong parent-child relationship that can affect this, and isn't the only route to respect. Respect, going both ways, is necessary for a strong relationship, though, and it is hard to develop without having well defined boundaries. It's worth considering how the relationship will develop if your child understands that he never has to listen to what she says unless you are around.
Lastly, this paradigm isn't uncommon, with the mother providing a safe, nurturing environment, and the father providing discipline and expectations. A lot of successful adults with strong relationship with both parents are thriving in the world today, and you wouldn't be the first family to go live this way. It isn't bad parenting, as long as there's a balance and the child's needs are met.
What you may find toxic to your family, however, is the feeling you have that you are doing work that should be a shared responsibility.
You either need to accept the work, and that your wife won't be doing it, or she needs to share that responsibility. If you continue to go along with this feeling of unfairness it will only force a wedge between you and her, and that will affect your child. While it's easy to say that she's the one who should change, it may well be that it's best if you change, and accept this responsibility. I don't know enough about you or your family to recommend that, I simply mention it as something you should seriously consider.