A parent who has more than one child with one or more partners: "Poly-what?"

With a single partner, either a man or a woman could be called a polypedonist, or more succinctly, polypedist:

From ancient Greek:

• πολύ (poly) = many
• παιδίον (paidion) = small child under training
• -ιστής (-istes) = active agent

One with many children.

Pedo- is the English root used in pediatrics and and pedophile, from the masculine παις, son, and the neuter παιδόν, including both male and female children. With the suffix -ist, the final o is normally dropped, but adding the n to form pedon- or adding an a to form paed- would both disambiguate from the ped of foot. From etymonline:

before vowels ped-, word-forming element meaning "boy, child," from Greek pedo-, comb. form of pais "boy, child," especially a son, from PIE root *peu- "small, little, few, young" (see few (adj.)). The British form paed- is better because it avoids confusion with ped-.

With multiple partners, either a man or a woman could be called a polykoinoteknist:

From ancient Greek:

• πολύ (poly) = many

• κοινός (koinos) = shared in common

• τέκνον (teknon) = offspring

• -ιστής (-istes) = active agent

One with many shared offspring.

The word παιδόν carried deeper connotations of ongoing nurture, while τέκνον generally referred to the physical offspring of any creature. If one really wanted to tweak the multi-partner arrangement, the designation could be altered to polykoitoteknist:

From ancient Greek:

• πολύ (poly) = many

• κοιτός (koitos) = marriage bed

• τέκνον (teknon) = offspring

• -ιστής (-istes) = active agent

One with many bedded offspring.

Though I am disgusted at the Greek-Latin lexical miscegenation, a parent of multiple children is polyprogenitive, from the Latin prōgeniēs, 'to beget'. From The Embodied Female:

Co-wives and surrogacy exist since Biblical times. For instance, resenting the four sons her fertile sister “gave” their common husband Jacob, barren Rachel gives him her maid Bilhah as surrogate, and adopts the two sons by this union. Emotions driving these transactions are unaccounted by Freudian chains of phallocentric symbolic equations slip-sliding between baby = penis or Darwinian notions of polyprogenitive male desire to spread his sperm. Neither can egg-exchange be entirely construed by Kleinian gynaecratic accounts of reparative urges to undo unconscious phantasies of raiding the archaic mother's envied fertile body for the babies within.

Technically, and although it doesn't fit the poly- mold requested, a person with more than one child would be multiprogenic (or multi-progenic). (The Greek equivalent, as Janus pointed out, is polytecnic, from τέκνον meaning 'child, son'.)

Similarly, a person who has children with multiple partners, according to social scientists, practices multipartnered fertility.

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In the simple case, a couple stayed married and had lots of kids (back when there was little else to do for fun, and no reliable means of contraception) there was no need to have a word for father or mother of a large family, as it was not unusual. Even now, this is the norm in some areas of society (Catholics and Mormons are stereotypical examples). A married woman who gets pregnant as often as possible is referred to as barefoot and pregnant or sometimes a baby factory or a breeder. These are all derogatory.

In traditional terms, when monogamy, cohabitation and marriage were assumed, serial partnerings resulted in step-fathers, step-mothers, half-brothers and half-sisters.

Outside of that paradigm of matrimony, nomenclature gets a bit cloudy (although half-brother and half-sister are still accurate). note: There's a certain amount of inherent asymmetry, in that you can't always prove who is the biological father (or used to couldn't) but there is rarely any doubt as to who is the biological mother.

If one woman has children by different fathers whom she did not marry, and does not live with, those men are baby daddies to their respective offspring, regardless of the number of children.

If one man fathers children by different mothers whom he did not marry, and does not live with, those women are baby mommas to his offspring, regardless of number of children.

If a man sires many children, each with a different baby momma, I would call him a seed-sower (i.e., he has been sowing wild oats.) Origin of phrase "sow wild oats"

If a woman bears many children, each conceived with a different baby daddy, you could call her a "serial baby momma". (That's derogatory, but not so derogatory as "slut". Of course, a [slut/ ho/ promiscuous woman/ sexually active female] might have lots of partners but no babies)

If a "baby daddy" disappears (whether by his choice or that of the baby momma), and plays no part in bringing up his offspring, he becomes what was once called an absentee father. Such a man is in legal terms a biological father, or colloquially, biological dad (sometimes more bluntly referred to as a sperm donor.) If the state keeps up with him and can keep him paying child support, he is a child-support payer. If not, he is a deadbeat dad.

If a man maintains a polygynous household, where multiple children born of several different wives are raised as his own, he is the patriarch of that family. Brigham Young, for example, was such a patriarch. (polygamy is no longer legal, nor sanctioned by the LDS church, but a few isolated splinter groups still practice it.)

I don't know of similar examples from polyandry, but I suppose by parallelism that the one mother in a polyandrous household with multiple husbands would be called the matriarch.

I would leverage polyphiloprogenitive, given in Mari-Lou's answer. But regarding the concern that it doesn't address the multiple reproductive partners, I would coin polyamoroprogenitive.

This combines polyamorous

The practice, state or ability of having more than one sexual loving relationship at the same time, with the full knowledge and consent of all partners involved. (Wikipedia)

and progenitive

capable of having offspring; reproductive (Dictionary.com)

It seems to me that the proffered polyphiloprogenitive satisfies both the "multiple partners" (poly philo) and the "multiple offspring" parts of the original question, without needing to stray into amor.

If one insists on purity-of-lexicon and is content to stay with Greek, then the term polyphiloteknic works perfectly. The k helps to distinguish the "offspring" part from the homophonic technic.