How can I pass extra parameters to UDFs in Spark SQL?

Just use a little bit of currying:

def convertDateFunc(resolution: DateResolutionType) = udf((x:String) => 
  SparkDateTimeConverter.convertDate(x, resolution))

and use it as follows:

case FieldDataType.Date => convertDateFunc(resolution(i))(allCols(i))

On a side note you should take a look at sql.functions.trunc and sql.functions.date_format. These should at least part of the job without using UDFs at all.

Note:

In Spark 2.2 or later you can use typedLit function:

import org.apache.spark.sql.functions.typedLit

which support a wider range of literals like Seq or Map.

You can create a literal Column to pass to a udf using the lit(...) function defined in org.apache.spark.sql.functions

For example:

val takeRight = udf((s: String, i: Int) => s.takeRight(i))
df.select(takeRight($"stringCol", lit(1)))