Mater dicit parcimoniam magno auxilio esse familiae.

Because this sentence uses indirect statement after dicit (says).

In Latin, after verbs of saying, thinking, knowing, hearing, and similar verbs, the reported statement is often expressed with:

• an accusative noun as the subject
• an infinitive verb

So:

• Mater dicit = Mother says
• parcimoniam ... esse = that thrift ... is

If this were a direct statement, you would expect the subject to be nominative:

• Parcimonia magno auxilio est familiae.
  = Thrift is a great help to the family.

When that statement is reported after dicit, parcimonia becomes parcimoniam, and est becomes esse.

For the same reason: this is an indirect statement.

In direct speech, Latin would use a normal finite verb:

• parcimonia ... est = thrift is ...

But in indirect statement, Latin changes the verb to an infinitive:

• parcimoniam ... esse = that thrift is ...

So esse is simply the infinitive form of to be.

Latin often does not use a separate word for that in this kind of sentence.

In English, we say:

• Mother says that thrift is a great help to the family.

In Latin, the idea of that is built into the accusative + infinitive construction:

• dicit parcimoniam ... esse

So there is no need for a separate word meaning that here.

Here familiae is dative singular: to the family.

The ending -ae can be confusing, because it can represent several forms in the first declension, including:

• genitive singular
• dative singular
• nominative plural

But in this sentence, the meaning and construction show that it is dative singular.

The phrase means that thrift is a great help to the family, so familiae names the person or group that receives the benefit.

Formally, auxilio could be either dative singular or ablative singular. But here it is dative singular.

The phrase auxilio esse means to be a help or more literally to be for a help. This is a standard Latin use of the dative with esse.

So:

• magno auxilio esse = to be of great help

The adjective magno agrees with auxilio, so it is also dative singular.

Yes. This is the double dative construction.

It contains two datives:

1. magno auxilio — the dative of purpose
   = for a great help, or more naturally a great help

2. familiae — the dative of reference/advantage
   = to the family

So literally, the phrase is something like:

• to be for a great help to the family

More natural English is:

• to be a great help to the family

This double dative is especially common with esse.

Latin word order is much more flexible than English word order because Latin uses case endings to show how words function in the sentence.

So Latin does not need to rely on word order as heavily as English does.

In this sentence:

• Mater is nominative, so it is the subject of dicit
• parcimoniam is accusative, so it is the subject of esse in indirect statement
• familiae and auxilio are understood from their case endings

That means the sentence could be rearranged without changing the basic meaning very much. For example, Latin could also say:

• Mater parcimoniam familiae magno auxilio esse dicit.

The chosen order may simply reflect style or emphasis.

The subject of dicit is Mater.

Mater is nominative singular and means mother.
Dicit is third person singular: she says.

So:

• Mater dicit = Mother says

After that, the sentence moves into the indirect statement.

In an ordinary clause, the subject is usually nominative. But in an indirect statement, Latin uses the accusative for the subject of the infinitive.

So inside the indirect statement:

• parcimoniam = subject of esse
• esse = infinitive verb

This is one of the most important Latin patterns to learn:

• verb of saying/thinking + accusative subject + infinitive

For example:

• Puella dicit se venire.
  = The girl says that she is coming.

Here se is accusative, but it is still the subject of venire.

The same thing is happening with parcimoniam ... esse.

A direct version would be:

• Parcimonia magno auxilio est familiae.
  = Thrift is a great help to the family.

Then, after Mater dicit, Latin changes it into indirect statement:

• Mater dicit parcimoniam magno auxilio esse familiae.

So the changes are:

• parcimonia → parcimoniam
• est → esse

That is the standard transformation from direct statement to indirect statement in Latin.