Hodie nobis cena paranda est.

This is the passive periphrastic, a very common Latin way to express necessity or obligation.

• paranda = a gerundive of parare (to prepare)
• est = is
• cena paranda est literally means something like dinner is-to-be-prepared

More natural English:

• dinner must be prepared
• dinner has to be prepared
• dinner needs to be prepared

So the sentence is not just saying that dinner is being prepared, but that it ought to be or has to be prepared.

Because cena is the subject of the passive periphrastic, so it stands in the nominative case.

Compare:

• cenam parare = to prepare dinner
  Here cenam is the direct object, so it is accusative.

But in:

• cena paranda est

the idea is passive: dinner must be prepared.
Now dinner is the grammatical subject, so Latin uses the nominative:

• cena = dinner as subject

This is one of the biggest adjustments for English speakers: the thing that must be done becomes the subject in Latin.

Paranda is the gerundive of parare.

A gerundive is a verbal adjective. In this construction, it usually carries the sense of:

• needing to be ...
• having to be ...
• fit to be ...

So:

• parandus, paranda, parandum = needing to be prepared

Here it appears as paranda because it has to agree with cena.

Since cena is:

• feminine
• singular
• nominative

the gerundive must also be:

• feminine
• singular
• nominative

So we get cena paranda est.

Nobis is dative plural of nos (we/us).

In the passive periphrastic, the person who has the obligation is usually put in the dative, often called the dative of agent.

So:

• nobis cena paranda est = literally dinner must be prepared by us
• more naturally: we have to prepare dinner

This is different from ordinary passive sentences, where you often get ab + ablative for the agent:

• cena a nobis paratur = dinner is being prepared by us

But with the passive periphrastic, Latin normally prefers:

• nobis

not

• a nobis

In this sentence, it mainly marks the people on whom the obligation falls.

So the best way to understand it is:

• it is necessary for us to prepare dinner
• we must prepare dinner

A very literal gloss might suggest by us, but the deeper idea is not simple agency in the ordinary passive sense. It is more like:

• as far as we are concerned
• for us
• our job is to prepare dinner

So if you translate too mechanically as dinner must be prepared by us, you get the structure, but the real force is closer to we have to prepare dinner.

Hodie means today.

It modifies the whole statement of obligation:

• Hodie nobis cena paranda est = Today we have to prepare dinner

Latin often puts time words like hodie near the beginning of the sentence, but word order is flexible. You could also see:

• Nobis hodie cena paranda est
• Cena hodie nobis paranda est

The meaning stays basically the same, though emphasis may shift slightly.

Yes. That is an excellent natural translation.

The Latin structure is literally closer to:

• today dinner must be prepared by us

but idiomatic English usually turns it into an active sentence:

• we must prepare dinner today
• we have to prepare dinner today
• today we need to prepare dinner

All of these capture the sense well.

Both can mean roughly the same thing:

• Hodie nobis cena paranda est
• Hodie debemus cenam parare

Both can be translated:

• Today we must prepare dinner

But the grammar is different.

Debemus cenam parare

• uses debeo
  • infinitive
• literally we ought to prepare dinner
• feels more straightforward to English speakers

Nobis cena paranda est

• uses the passive periphrastic
• literally dinner is to-be-prepared by us
• often sounds a bit more formal, objective, or impersonal in Latin

Latin likes the passive periphrastic very much, especially when emphasizing the necessity of the action rather than the personal decision of the doer.

No.

• paratur = is being prepared / is prepared
• paranda est = must be prepared / needs to be prepared

So:

• cena paratur = dinner is in the process of being prepared
• cena paranda est = dinner still needs to be prepared, or there is an obligation to prepare it

This is an important distinction:

• paratur describes an action or state
• paranda est expresses necessity

Grammatically, the subject is cena.

That can feel strange to an English speaker, because in the natural English translation we often say:

• we must prepare dinner

where we is the subject.

But in the Latin sentence:

• cena = the grammatical subject
• nobis = the person who bears the obligation

So Latin organizes the thought differently:

• dinner is the thing that needs doing
• we are the people responsible for doing it

Yes. Latin word order is quite flexible because the endings show the grammatical relationships.

So all of these are possible in principle:

• Hodie nobis cena paranda est
• Nobis hodie cena paranda est
• Cena nobis hodie paranda est
• Paranda est nobis hodie cena

They do not all sound equally neutral, but they are understandable because:

• cena is nominative
• nobis is dative
• paranda agrees with cena

The given order is a very natural one:

• Hodie sets the time first
• nobis introduces who is concerned
• cena paranda est gives the main obligation

In normal prose, yes, you expect est here.

The passive periphrastic is built from:

• gerundive + form of esse

So:

• paranda est
• not just paranda

In some special contexts, especially poetry or very compressed style, a form of esse can be omitted, but for a learner, you should think of est as an essential part of the construction.

Because it agrees with cena.

Cena is:

• feminine
• singular
• nominative

So the gerundive must match it exactly:

• masculine singular nominative: parandus
• feminine singular nominative: paranda
• neuter singular nominative: parandum

Since the noun is cena, the correct form is paranda.

This agreement is just like ordinary adjective agreement in Latin. The gerundive behaves like an adjective, even though it comes from a verb.