You can use the --parameter-overrides switch. The syntax is quite long winded, as below:

sam local start-api --parameter-overrides ParameterKey=Key1,ParameterValue=value1 ParameterKey=Key2,ParameterValue=value2

That is, you need to specify the key and value of each pair with comma separation.

And then each pair is separated with a space.

From sam local start-api --help:

  --parameter-overrides       Optional. A string that contains
                              CloudFormation parameter overrides encoded
                              as key=value pairs. Use the same format as
                              the AWS CLI, e.g. 'ParameterKey=KeyPairName,
                              ParameterValue=MyKey ParameterKey=InstanceTy
                              pe,ParameterValue=t1.micro'

You can use the --parameter-overrides in sam deploy just like in aws cloudformation deploy with a small change:

Before:

sam deploy --template-file packaged.yaml --stack-name example-lambda --capabilities CAPABILITY_IAM --parameter-overrides ParameterKey=SourceS3Bucket ParameterValue=test-data-111

After:

sam deploy --template-file packaged.yaml --stack-name example-lambda --capabilities CAPABILITY_IAM --parameter-overrides SourceS3Bucket=test-data-111

Noticeable change: ParameterKey, ParameterValue does not need to be explicitly specified in sam deploy. Helps me in local testing.

Hope it helps. :)

It seems you can also use the -n or --env-vars parameter to pass environment variables in a JSON file to your functions. See the docs: Test Your Serverless Applications Locally Using SAM CLI (Public Beta)

In short, your JSON file would look like (example copied from documentation):

{
  "MyFunction1": {
    "TABLE_NAME": "localtable",
    "BUCKET_NAME": "testBucket"
  },
  "MyFunction2": {
    "TABLE_NAME": "localtable",
    "STAGE": "dev"
  },
}

And then you can do:

 $ sam local start-api --env-vars env.json

This is specifically for environment variables for your lambda functions though, so it may not be entirely what you're after?