Engels' essay, "On Authority", for long has been cited as one of the so-called "refutation" of anarchism and anti-authoritarianism by Marxists. Many excellent works have been done to critique this essay, namely section H.4 in The Anarchist FAQ as well as the essay titled "On Authority" Revisited. However, something not particularly touched upon by these two works are the root of the semantical issue of the essay, and how this relates to modern day anarchism.
Engels and the two anarchist views on authority
Anarchists historically have viewed authority in 2 different manner, primarily stemming from operating under different definitions. The first view defines authority broadly to include things such as expertise and force, and thus rejects only "unjustified" authority. Such was the anarchism of Bakunin and Chomsky: Bakunin stated that he "bow[s] before the authority of special men because it is imposed on me by my own reason" (Bakunin, What is Authority?) and Chomsky stating that "Authority, unless justified, is inherently illegitimate and the burden of proof is on those in authority." The later view does not define authority vaguely, instead defining authority as based on domination and hierarchization.
The issue that lies with Engels' critique is that he falsely conflate these two incompatible views: he takes the definition of the former (which encompasses expertise and force) and combine such with the opposition to all authorities from the later. This creates a strawman that represents no anarchists: for anarchists have no qualms with neither the "authority of the railway employees over the passengers" in regards to how trains are ran nor the "authority" of a coordinator in large projects and industries: under the first view such position is justified, and in the later not authorities at all.
The problem with Bakunin/Chomsky's view on authority
Let us continue thus to the problems with Bakunin and Chomsky's understanding of authority. Given Chomsky's popularity within some anarchist circles and more generally as a leading leftist intellectual, he is often cited regarding anarchism and authority, hence leading to this view being widespread among modern day anarchists. While not ahistorical (for his operating definition stemmed from Bakunin), this view of authority can be harmful towards a comprehensive understanding of anarchism.
The first and most notable issue is how every past movements proclaim to be for "justified authority". Monarchists claim the king possess an authority justified by the will of God or inheritance, while liberals claim that capitalists have justified authority from their claim of ownership of land and instruments obtained through "voluntary" exchange. This understanding thus fails to meaningfully and easily distinguish anarchism's opposition to authority from that of liberals and other statists. This is not a projected or imaginary issue at all, one can see it all too clear in Chomsky's own writings and statements: he argued that "the strategy of the sincere anarchists must be to defend certain State institutions against the assaults they undergo" (Quoted by Claude Guillon in "The Chomsky effect or anarchism of the State") and defended supporting figures such as Hillary Clinton as the "lesser evil". In doing so, he proposes an un-anarchistic solution: instead of developing genuine anti-authoritarian oppositions through mutual aid and popular resistance, he wishes to redirect effort to the "lesser evil" in the state.
The second issue is how such framing can (and does) lead to confusion about what "authority" is justified and what is not. Often one sees anarchists claiming that "authority" is justified in relationships such as parent-child or in the organization of industry, but where is the line drawn as to when these relationships and mode of organization become "unjustified"? Where does the "authority" the parent has to discipline their child end and where does abuse begin? The answer remains unclear, with "anarchists" of old like Proudhon proclaiming support for the familial authority, a sentiment surprisingly echoed by many modern day anarchists .
A clear anarchist view of authority
Instead of the view riddled with problems seen above, anarchists should instead opt for the view where "authority" is defined by domination and hierarchization, and as such all authorities are opposed. "Domination" is defined as having the capacity to dictate others' actions without constraint (for a more detailed discussion on domination see Christopher McCammon's 2015 Domination: A Rethinking and article on SEP titled "Domination"). "Hierarchies" are consequently defined as a system of systemic domination. Indeed, with such a view, things become much more clear. There is no longer any possible justification of kings and capitalists: for it is clear that the relationship between they and their subjects is one based on hierarchies and domination.
The line in parenting and organization also becomes clear: so long as the parent treat the child as an equal whom they are supporting rather than a subject which they dictate, an anarchist would not oppose such. However when the parent treats the child as a subject of their own control (such as by disregarding the child's will and instead projecting what they think is good for the child unto them), the relationship becomes one of hierarchization and domination, and as such should be opposed by anarchists. In the words of Peter Gelderloos, "children are given both autonomy and encouragement to learn, along with every possibility to self-actualize." (Gelderloos, Defying Power). Similarly, an anarchistic coordinator has no authority over other workers but instead is one of them. When a coordinator he asserts himself over the will of others and seek to control workers, this relationship ceases to be anarchistic and turn authoritarian.