The issue of conscription is quite the political football, prone to be used for political posturing rather than genuine conviction. During the early years of the Iraq War, for instance, some Democrats proposed bringing back the draft, then voted against it, just so they could engage in technically true, but baldly dishonest, scare-mongering about the draft coming back. And @Eremi responded to @Philipp 's discussion of the Draft America's Daughters Act of 2016 introduced by Duncan Hunter by claiming that Hunter intended the bill for rhetorical effect, rather than truly being motivated by a desire to draft women. Thus, simply noting that a bill was proposed does not settle the issue.
However, some examples that appear to be made in earnest are discussed here:
“This is absolutely unfair,” Rep. Mike Coffman, a veteran of both Iraq wars, says of the sanctions. Coffman, a Republican from Colorado, is among the most outspoken critics of the system. Earlier this year, he co-sponsored a bipartisan bill that would abolish the $23 million agency, suspending registration – and sanctions — except by executive order in a time of national emergency.
This would partially address the inequality by removing the current requirement that men register. Going the other way and addressing the inequality by requiring women to register:
One of them, he notes, belongs to Democratic Rep. Charles Rangel of New York, who earlier this year introduced two bills related to the draft. The first would require women to register with the Selective Service. The second calls for all citizens and residents between 18 and 25 to perform two years of military or community service and would reinstate the draft only when a clear threat to the nation is present and Congress formally has declared war or the president proclaims a national emergency.
@Venture2099 has made a big deal out of the fact that there is no current draft. Note that the original wording of the question was "Are there any USA politicians who actively pursuit equal rights when it comes to conscription?" and the question has been modified to, as of this writing, "Are there any USA politicians who actively pursuit equal rights when it comes to potential conscription by requiring women to register too?" I think that charity requires us give reasonable leeway to interpret questions in a way that avoids making them make false claims. "comes to" can be interpreted as "on matters related to", and the Selective Service is related to conscription, and it would be silly to try to analyze the SS without reference to conscription. Furthermore, SS rules are having real, present effects on men, and is not merely a "potential" or "theoretical" issue. As the article I linked to mentions, men who do not register can be denied benefits such as certain jobs, job training, educational financial aid, and citizenship. Note that the requirement to register runs from age 18 to 26, and while some of these restrictions apply only to that age range, in some cases, a man who passes the age of 26 without registering is forever barred from certain benefits, as they are no longer able to register (that is, some restrictions are of the form of "someone who can register but hasn't is barred", while others are of the form of "someone who was supposed to register but hasn't is barred"; in the latter case, once a person passes the registration age, they are forever ineligible). It is safe to say that men have lost out on billions of dollars of benefits due certain restrictions, and quibbling about exactly what wording should be used when discussing the relationship between these restrictions and the draft is, (especially when those quibbles are used to completely dismiss the question rather than just assert that it is poorly worded), in my mind, somewhat bad taste.