Political parties in the US are similar to "Political Action Committees" (PACs), and are organized as non-profit organizations under the IRC Sec. 527.
As such their contributions or donations income is not taxable, and anyone can give them money as a non-taxable (and non-deductible) gift.
The donation limits are specifically for the candidates themselves. An individual cannot give more than $X to the candidate (or their campaign), but they can give whatever they want to any PACs. The PACs (including the DNC/RNC/State or county level committees/etc) will then provide support to their candidates. The "Citizens United" precedent cited in the other answer removed any limitations on what PACs can do, as long as they don't directly coordinate their actions with the candidate themselves.
Political party committees are recognized explicitly in FECA and have somewhat fewer restrictions than (other) PACs, one major is that they can coordinate with candidates, and move money more easily within the party organization (PACs are limited how much they can transfer to other PACs).
In which case, it seems also like "bribery" (the agency will pay for the politician, in return the politician is their employee that has to make them money and obey their direction)?
This is generally how it works in the US, and is not considered "bribery". Although in some cases people did run afoul of the very few laws remaining that tried to shield politicians from such influence (one notable recent example is a certain criminal defendant's "charitable" arm donating to the prosecutor tasked with charging that said defendant).