Regarding Political Relations
The relation between capitalism and politics is multifold. First, political systems in part operate to ameliorate problems and defend prerogatives of any economy they accompany, capitalism included. This leads to pictures like that provided above. Second, capitalism establishes all kinds of power differentials that in turn impact who can affect political outcomes. These range from generating gigantic corporate centers of power to unequally empowering individuals, as well.
To make a very long story short, consider military spending. On the one hand, it provides armed power to defend and extend the sway of the owning class, domestically and abroad. But that isn't the whole of its appeal, because in fact most military spending is waste and redundancy, etc., as everyone knows, and therefore can't be explained on the basis of its material output.
Military spending occurs not only in pursuit of military products, that is, but even when there are no desired products to be had. Reasons generally offered are that it primes the pump of capitalist production. It generates profits for massive owners. It provides jobs.
But there is a problem. Building schools, hospitals, public transport systems, parks, and public and low cost housing would also prime the pump of capitalist production, generate profits for massive owners, and provide jobs -- and could do all three at an even larger scale than military spending, for that matter. So why does the political system opt for even redundant and just plain insanely wasteful military spedning instead of productive social spending?
The answer is that military production doesn't disturb the balance of power among contending classes to the advantage of the poorer and weaker ones. In contrast, social spending does, by educating, insuring against unemployment, stabilizing life and raising expectations, and so on. So, the polity organizes collective expenditures on guns and not on schooling, housing, and food, precisely due to the social effects of the latter, which are not wanted due to their hierarchy-destabilizing implications. This is, of course a remarkable statement about both the economy and the polity in capitalist societies.
Regarding Political Relations
In a parecon, the implications of the economy for the polity are substantial. First, the economy produces an inclination toward and a familiarity with self managment. The economy is a school for participation, so to speak, which makes it hard to imagine a parallel polity which denies participation to its citizens. Likewise, a parecon gives no economic actors any material or social means to have any more say or even to seek any more say than any other actors.
The economy removes from the polity much of its focus in other societies. There is no longer need to defend the prerogatives of a ruling class nor to punish the violations of dissenters from ruling class norms. There is no ruling class. There is no need to clean up economic messes, nor to ameliorate economic injustices.
Political functions such as adjudication of disputes, legislation of shared norms, and implementation of collective projects and programs still exist. We still have (albeit new) institutions for handling social violations and crimes, disputed possibilities, lawful norms, and so on. What is changed is that however the polity operates, the economy does not subvert democratic possibilities by enriching and empowering a relative few people and impoverishing and debilitating many others. And to the extent the polity engages in production, consumption, and allocation -- it does so via parecon's balanced job complexes, councils, and participatory planning.
It is also true, however, that an authoritarian polity might well represent a problem for a parecon -- producing citizens expecting to order and be ordered -- who would not fit well in parecon's settings. Thus, in addition to not subverting political participation and self management, by the training it affords, the aspirations it arouses, and the needs it has for a participatory workforce, a parecon creates great pressure that an accompanying polity diminish authority to a minimum and elevate self management to a maximium.