Learn Finnish: Pojat eivät ottaneet laskuvarjoa. - The boys did not take parachutes.

Yes, laskuvarjoa is the partitive singular form. Finnish often uses the singular when a group of people each interact with exactly one item. Since each boy would only take one parachute, Finnish uses the singular. English naturally prefers the plural 'parachutes' here.

If the sentence were positive, would the object still be in the partitive case?

No. If the boys did take parachutes, the sentence would be Pojat ottivat laskuvarjon, using the genitive singular to show a fully completed action. Because our sentence is negative (eivät ottaneet), the object is forced into the partitive case (laskuvarjoa).

Why is the main verb ottaneet and not ottanut?

In negative past tense sentences, the main verb takes a participle form that must agree with the subject in number. Because the subject Pojat is plural, the verb takes the plural participle form ottaneet. If the subject were singular (e.g., Poika), it would be ottanut.

What would it mean if I used the plural partitive laskuvarjoja instead?

If you said Pojat eivät ottaneet laskuvarjoja, it would imply that each boy was expected to take multiple parachutes, or that they were dealing with a general, undefined pile. Using the singular laskuvarjoa strictly highlights the 'one parachute per boy' rule.