In Spain we use "madre" and "padre" when you refer to your parents (or somebody else's parents).
Mi padre le ha regalado a mi madre un collar por su cumpleaños.
Tu padre está un poco loco Juan.
El padre de Luis es muy estricto; la madre un poco menos.
We call then "papá" and "mamá" when addressing directly to them or when talking about them with our siblings.
Papá, ¿qué hay para cenar?
Mamá, papá quiere saber dónde has puesto sus revistas.
(Talking to my brother or sister) Ayer papá y mamá me echaron la bronca por llegar tarde.
There is also "papi" and "mami" (daddy, mamma/mammy), but they sound so cheesy that only small kids would use it.
I noticed that here in the States, where I live, people would say "My dad...". In Spain we use "Mi padre...". I think that some Latin American countries would use "Mi papá..." but in Spain saying "Mi papá...", unless you are a toddler, sounds like if you were saying "My daddy..." in English (little cultural differences).
I have heard many speakers from different Latin American countries use "papá" and "mamá" to refer to their parents to others, but I'm not sure if this is due to an influence of being in an English speaking country where they constantly hear "My dad..." of if it is also to norm in their own countries.
You could also use "papaíto" and even "mamaíta", but as with "papi" and "mami" only small children (or cheesy young adults) would use these terms.