Modo numerus par est, modo impar est.

Here modo ... modo is a correlative adverb pair. It means something like:

• now ... now
• at one time ... at another
• more naturally in English, sometimes ... sometimes

So the sentence is not using modo in its other common senses such as only or just now. The repetition is the clue: modo ... modo sets up an alternation.

Numerus is the subject of the sentence. It is nominative singular and means number.

So in the first clause:

• numerus = the number
• par est = is even

Altogether: the number is even.

Yes. Par and impar are adjectives.

In this sentence they are predicate adjectives, meaning they describe the subject after a form of to be:

• numerus par est = the number is even
• numerus impar est = the number is odd

They agree with numerus in case and number: both are nominative singular here.

Because they are not first-and-second-declension adjectives. They belong to the third declension.

Their dictionary forms are:

• par, paris = equal, even
• impar, imparis = unequal, odd

So par is already the correct nominative singular form, and impar is too. There is no need for endings like -us here.

Latin often leaves out a word when it is easily understood from the context. In the second clause, numerus is simply understood:

• modo numerus par est, modo impar est
• literally: now the number is even, now [the number] is odd

English does this too sometimes, but Latin does it very comfortably.

Yes, Latin can sometimes omit a repeated est if the meaning is clear. But here repeating est makes the two halves nicely balanced:

• modo numerus par est
• modo impar est

That repetition gives the sentence a clear rhythm and contrast. It is very natural Latin.

Because modo ... modo already does the linking work. It sets up the contrast or alternation by itself.

So Latin does not need an extra conjunction such as et or sed here. The repeated modo is enough to show:

• one situation at one time
• another situation at another time

Yes. Latin word order is more flexible than English word order because the endings show the grammar.

For example, the following would still be understandable:

• Modo par numerus est, modo impar est.
• Numerus modo par est, modo impar est.

But the given version is very straightforward and elegant. Putting modo first in each clause clearly marks the alternation, and numerus par est is an easy, natural order for a learner to recognize.

No. A single modo can mean several things depending on context, for example:

• only
• just now
• provided that

But when you see modo ... modo, you should usually think of the repeated pair meaning now ... now or sometimes ... sometimes. So the repetition is important.

Yes. In mathematical or numerical contexts:

• par = even
• impar = odd

More literally, par has the idea of equal or matched, and impar means unequal or unmatched. From that basic idea, they also become the standard terms for even and odd numbers.

It is basically two simple clauses joined in a balanced pattern:

1. modo numerus par est
2. modo impar est

Each clause has:

• an adverb: modo
• a subject, expressed or understood: numerus
• a predicate adjective: par / impar
• the verb est

So it is a neat example of a copular sentence with sum plus a predicate adjective, repeated in a correlative structure.