Which is grammatically correct? Are or is?

The plural form “are” is correct here.

Since two things are dropping, both “size” and “cost”, then those things “are” dropping.

Thanks @CWOTUS

Eliminate “of parts,” which is a modifier and try the simplified version; “The size and cost are dropping.”

However, I am not sure what “the size of parts is dropping” would mean. Those particular parts are being made in smaller sizes ?

In English make sure verbs and subjects agree – singular subjects have singular verbs and plural subjects have plural verbs. Sentences with two subjects joined by and take a plural verb, even if each of the two subjects is singular.

You could rephrase it. ‘Parts are getting smaller and cheaper” for example.

The size of parts is dropping. (singular)
The and cost of parts are dropping. (plural)

Parts and cost are getting cheaper.
Rules:
http://www.grammarbook.com/grammar/subjectverbagree.asp