I've read where few top politicians in the country recommended we switched over to parliamentary system, and on different occasions I can't help but notice that members here find it quite incomprehensible.

So I'll go on and clarify few things in the processes so you can have your own opinion on what you think Nigeria needs.

Parliamentary system of government is a very broad topic because there are many types practised across the globe- it appears most countries customize theirs by how it can be very effective and benefit them most.

Since the proponents were mostly concerned about accountability and lower cost of governance, I'll rush into the Multiparty Parliamentary Republic system- where the head of government is also the head of state. And for conciseness and readability, I'll bring this to Nigeria context instantly by using the presidential system (you already know) to simulate it.

Process

If we decide to switch over to a parliamentary system, there will be an amended constitution of course, where the office of the president and commander-in-chief will be dismissed.
Therefore when we hold our general elections, we will only vote individuals from our respective senatorial zones into the senate and the party with the highest number of seats forms the government and elects a prime minister, who will then be in charge of executive (ie the president) and also answerable to the senate(parliament) at the same time. The prime minister appoints his executives(ie cabinet members or ministers) and resumes duty as both the president and senate president(Nigeria context)

While the party with the second highest seats forms an official opposition which will be recognized by the constitution.
Then depending on what Nigeria really wants, the lower house(House of Reps) can be customized in such a way they they will be relevant in the upper house's(Senate's) policies, or if not, the lower house can be dismissed too by the constitution(ie to achieve more reduction in cost of governance)

Arguments for Parliamentary System:

Accountability:
Parliamentary system has been unarguably the most transparent system of government. The theory of collective responsibility ensures that the Prime minister and his cabinet takes all glory and at same time responsible for all failures with no excuses. The government can always be dissolved if the parliament passes a 'vote of no confidence' and the majority party will carefully re-elect a new head(prime minister) to ensure their party's continuity (ie highest number of seats in next election)

No Tribalism:
There will hardly be a case of 97% and 5%, since all parties must prove their governance ability in all states, if not for anything, for the sake of winning the highest number of seats since each state must produce 3 senators. If a party messes up with a region, 15 or 18 of their senators may not be returning to the parliament after elections(ie 3 senators multiplied by number of states in the geo-political zone). Truly, this will aid faster development of all regions cos obviously, every region matters. One senator less can force a party into the opposition.

Responsible governance & less likely to become dictatorial:
In the parliament it will be a serious business because unpopular policies will be openly challenged and criticized by the opposition party which is officially recognized by the constitution unlike in the presidential system that they're only 'noise makers' and 'wailers' The opposition can always take advantage of any neglected state to win more senate seats, so would the other minor parties.

Corruption is easily checked:
If $44 million goes missing in NIA vaults, there will be no need of setting up committees and sending delegates to anybody since the prime minister and his cabinet are right there in the parliament together with other power-hungry parliamentarians. Even the prime minister's very own party poses a threat to him cos surely some other members wants to be prime minister themselves. Moreover only the prime minister's LGA is enough to deal with him in upcoming elections(if he loses in his own LGA, he may not stand a chance emerging from that zone again)

Brings government closer to the people & Laws are enacted faster:
Like when the house was against police brutality, they wouldn't have to set up a committee that will investigate and send delegates to the executive, rather they would start the reform process right there in the parliament and pass it into law(in this case the representatives must represent their people or rather prepare on how to face them when he/she gets to the village next time, no excuses)

At last, the very backbone of the parliamentary system is a free and fair election(which might be easier to achieve cos it's an average of 10 LGAs within a state that will be producing one candidate)


Arguments Against Parliamentary System:

The most popular disadvantage is it's violation of Separation of Power:
Power is fused between the legislative and executive which largely questions the principles of checks and balances. But then there's official opposition, and if free and fair elections is achieved, then there's the electorates too.

Struggles between the ruling party and the opposition:
Our power-hungry opposition may over criticize the government and such destructive criticism may somehow delay the progress of national development.

Strict adherence to party discipline:
This implies that a member cannot, for example; vote against his/her own party's policy, otherwise must follow due process of cross-carpeting(defecting) as stipulated by the constitution if he/she has lost confidence in the party.
But this right here is not uncommon as practised in presidential system except that there's no strict law compelling party members to always agree.

There would be more major disadvantages if the head of government is not the head of state.


Few countries that practise Parliamentary system:
UK, Australia, Canada, New Zealand, Japan, Austria, Denmark, France, Belgium, Finland, Germany, Israel, Malaysia, Singapore, Thailand, Pakistan, Indian, Bahamas, Barbados, South Africa, Ethiopia, Hungary, Ireland, Italy, Luxembourg, Malta, Netherlands, Norway, Spain, Switzerland, Sweden, Turkey, etc

Few countries that practise Presidential system:
USA, Angola, Benin, Nigeria, Burundi, Cyprus, Malawi, Indonesia, Philippines, Zimbabwe, Zambia, Afghanistan, Chile, Guatemala, Iran, Liberia, El Salvador, Colombia, Kenya, Palau, Turkmenistan, Sudan, Paraguay, Mexico, Comoros, Maldives, Costa Rica, Nicaragua etc

Presidential system relies hugely on patriotism to be successful.
As you can see from the list, apart from the great giant USA, no other first world country practises the presidential system. Most of the other countries appear to be struggling.

The system of an individual wielding absolute power can be too complicated for young and multi-ethnic nations to handle.

NB: This is just the basics of parliamentary system, like I said every country modifies how it suits the population

Know what you prefer today.

Questions are welcome too

We already had this, but some misguided bigots weren't content with it. They wanted to rule everywhere and everyone. The so called One-Nigerianists!

If a country want to develop and shun tribalism the answer is parliamentary system of government take instance Switzerland and South Africa you hardly know they were of different tribe.

dignity33:
If a country want to develop and shun tribalism the answer is parliamentary system of government take instance Switzerland and South Africa you hardly know they were of different tribe.

Very correct. South Africa has 10 major tribes, yet you hardly know about this and assume all of them as one.
The simple truth is that each tribe controls equal amount of Power and also each of the tribes needs and must always seek out to the others no matter how few they are, cos losing them to the opposition might be dangerous for the party. That way you see optimum cohesion in government and progress amongst the governed. Nothing like 95% and 5% in the system